Analysis 1 & 2 — Recap

Typeset recap for ETH ITET Analysis 1 and Analysis 2: definitions, when-to-use maps, worked examples, and exercises. Read straight through or jump to a chapter below.

Analysis 1 — core topics #

1. Foundations — logic, maps, sup/inf, induction; the language for every proof later.
2. Sequences — convergence, Cauchy sequences, and a limit triage map.
3. Series — convergence tests, power series, and when each test fits.
4. Functions — continuity, compactness, and the theorems behind existence proofs.
5. Sequences of functions — pointwise vs uniform convergence; when swapping limits is safe.
6. Derivatives — rules, mean value theorem, Taylor, and Newton.
7. ODEs — first-order and linear systems; homogeneous + particular workflows.
8. Differential calculus in $\mathbb{R}^n$ — partials, Jacobian, chain rule, implicit functions.
9. Extrema — free and constrained optimization; Hessian and Lagrange.
10. Integration (Ch. 10) — Riemann, FTC, techniques, line integrals, and solids of revolution.

Analysis 2 — core topics #

11. Integration (Ch. 11) — Jacobians, Green, Gauss, and volume and flux in $\mathbb{R}^n$.
12. Potential fields — gradients as potentials; path independence and holes.
13. Vector analysis — grad, div, rot, and the Green–Stokes–Gauss ladder.
14. Topology — open sets, compactness, simply connected domains.
15. Reference formulas — algebra, trig, linear algebra, geometry, and $dV$ factors at a glance.
16. Complex numbers — polar form, holomorphic functions, Cauchy, and residues.
17. Additional integrals — substitution templates beyond the standard table.
18. Fourier series — orthogonality, coefficients, convergence, Parseval.

Chapter 1 — Foundations (Grundlagen) #

Logic, sets, maps, bounds on $\mathbb{R}$, summation tools, induction, partial fractions, and inequalities — the language every later chapter uses.

Story of the chapter: before limits and derivatives you need precise statements (“for all”, “there exists”), correct negations, control of sup/inf on $\mathbb{R}$, and algebraic templates (telescoping sums, partial fractions) that reappear in Ch 2, Ch 3, and Ch 10.

Prerequisites: secondary-school algebra.

Foundations ladder (how the pieces fit)

1.1 Quantifiers #

1.1.1 Overview #

Quantifiers

Logic connectives

Sets (notation)

Typical mistake Confusing $\forall$ with $\exists$: “for all” is a strong claim (one counterexample kills it); “there exists” is weak (one witness is enough).

1.1.2 Negation rules #

When to use

• Rewriting proof goals (“show not convergent” $\Leftrightarrow$ “$\exists\varepsilon$ …”)
• Checking whether a statement is false

Proof moves (exam level): direct proof; contrapositive ($A\Rightarrow B$ via $\neg B\Rightarrow\neg A$); proof by contradiction (assume $\neg B$, derive absurdity).

1.2 Maps (Abbildungen) #

1.2.1 Surjectivity #

$f:A\to B$ is surjective iff

When to use

• Showing $f$ hits every value in the codomain; constructing preimages

Typical mistake Checking only a few sample $y$ values instead of all of $B$.

1.2.2 Injectivity #

$f:A\to B$ is injective iff

(equivalently: $x_1\ne x_2\Rightarrow f(x_1)\ne f(x_2)$).

Useful test (Ch 6): if $f$ is differentiable and $f'>0$ everywhere on an interval, then $f$ is strictly increasing and hence injective there.

1.2.3 Bijectivity and inverse #

$f$ is bijective iff it is injective and surjective.

If $f:A\to B$ is bijective, the inverse $f^{-1}:B\to A$ satisfies

Typical mistake Defining $f^{-1}$ when $f$ is not bijective (e.g. $e^x:\mathbb{R}\to\mathbb{R}$ — restrict codomain to $(0,\infty)$ first).

1.2.4 Composition example #

If $f:X\to Y$ and $g:Y\to X$ satisfy $g\circ f=\mathrm{id}_X$, then $f$ is injective and $g$ is surjective.

1.2.5 Composition rules #

For $f:A\to B$ and $g:B\to C$:

Typical mistake Assuming $g\circ f$ surjective when only $f$ is surjective (need $g$ surjective on all of $B$).

1.3 Supremum and infimum #

• Supremum $s=\sup A$: $a\le s$ for all $a\in A$, and if $o$ is any upper bound then $s\le o$.
• Infimum $i=\inf A$: $a\ge i$ for all $a\in A$, and if $m$ is any lower bound then $i\ge m$.

Key identities

Completeness of $\mathbb{R}$: every non-empty set bounded above has a supremum in $\mathbb{R}$.

Link to Ch 2: a monotone bounded sequence converges to its sup/inf.

$\max$ vs $\sup$: on a closed bounded set $K$, a continuous function attains $\max$ and $\min$ on $K$ (Ch 4 EVT); for open sets, only sup/inf need exist.

Typical mistake Assuming $\sup A\in A$ (e.g. $\sup(0,1)=1$ but $1\notin(0,1)$).

1.4 Triangle inequality #

For $x,y\in\mathbb{R}$ (and in $\mathbb{C}$ with modulus):

Use when

• Estimating integrals, limits, and norms in $\mathbb{R}^n$ (Ch 8, Ch 11)

Typical mistake Writing $\lvert x+y\rvert=\lvert x\rvert+\lvert y\rvert$ without checking signs (equality in $\mathbb{R}$ only when $x,y\ge 0$ or both $\le 0$).

1.5 Manipulation of sums and products #

1.5.1 Telescoping sums #

Index shift: $\displaystyle\sum_{k=m}^{n} a_k=\sum_{j=m}^{n} a_j$ — rename the index when aligning with a formula.

1.5.2 Product telescoping #

1.5.3 Standard finite sums (use or prove by §1.6) #

Link to Ch 3: telescoping and these sums appear in partial-sum arguments.

1.6 Complete induction #

To prove $A(n)$ for all $n\ge n_0$, $n\in\mathbb{N}$:

1. Base: verify $A(n_0)$.
2. Inductive hypothesis: assume $A(n)$ for some $n\ge n_0$.
3. Step: prove $A(n+1)$ using the hypothesis.

When to use

• Sums, divisibility, inequalities with $n$

Typical mistake Using $A(n+1)$ as if already proved without stating the inductive hypothesis; trying to induct a false universal statement (see Ex C1).

1.7 Partial fraction decomposition #

Workflow (exam recipe)

1. If $\deg P\ge\deg Q$, perform polynomial division: $\dfrac{P}{Q}=S+\dfrac{R}{Q}$ with $\deg R<\deg Q$.
2. Factor $Q$; find all zeros.
3. Assign unknown constants to each partial template.
4. Clear denominators or compare coefficients.
5. Verify by recombining.

Real simple pole at $x_0$:

Real $r$-fold pole:

Irreducible quadratic $ax^2+bx+c$ with discriminant $b^2-4ac<0$ (no real roots):

Complex $r$-fold root: same template with linear numerators (coefficients may be complex; recombine to real fractions when possible).

1.8 Method map (quick reference) #

1.9 Foundations — Exercises #

A) Direct application #

A1 Negate: $\forall x\in\mathbb{R}\,\exists y\in\mathbb{R}:\ x+y=0$.

A2 Is $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^3$ injective? surjective? bijective?

A3 Compute $\displaystyle\sum_{k=1}^{n}\frac{1}{4k(k+1)}$.

A4 Find $\sup$ and $\inf$ of $A=\{x\in\mathbb{Q}: x^2<2\}$ in $\mathbb{R}$.

B) Multi-step #

B1 Decompose $\dfrac{x^2+1}{x^3-x}$ into partial fractions.

B2 Prove by induction: $2^n\ge n+1$ for all $n\in\mathbb{N}$.

B3 Prove: if $f,g$ are injective, then $g\circ f$ is injective (§1.2.5).

C) Mixed exam-style #

C1 Let $A(n)$ be “$n^2+n+41$ is prime”. Test small $n$, then discuss why induction of a false universal claim fails (find $n$ with $A(n)$ false).

C2 Decompose $\dfrac{2x+3}{(x-1)^2(x+2)}$ and integrate (preview Ch 10).

Answers (A) #

A1 $\exists x\in\mathbb{R}\,\forall y\in\mathbb{R}:\ x+y\ne 0$.

A2 Injective yes ($x_1^3=x_2^3\Rightarrow x_1=x_2$); surjective yes (every $y$ has root $y^{1/3}$); bijective yes.

A3 $\dfrac{1}{4}\bigl(1-\dfrac{1}{n+1}\bigr)=\dfrac{n}{4(n+1)}$.

A4 $\sup A=\sqrt{2}$, $\inf A=-\sqrt{2}$ (neither lies in $A$).

Answers (B) #

B1 $\dfrac{x^2+1}{x^3-x}=\dfrac{-1}{x}+\dfrac{1}{x-1}+\dfrac{1}{x+1}$ (see worked solution below).

B2 Base $n=1$: $2\ge2$. Step: $2^{n+1}=2\cdot 2^n\ge 2(n+1)=2n+2\ge n+2$ for $n\ge 1$.

B3 $g(f(x_1))=g(f(x_2))\Rightarrow f(x_1)=f(x_2)\Rightarrow x_1=x_2$ by injectivity of $f$ then $g$.

Answers (C) #

C1 $A(n)$ holds for many small $n$ (e.g. $n=1,\ldots,40$) but fails at $n=41$ since $41^2+41+41=41\cdot 43$ is not prime. Induction cannot prove a false universal statement.

C2 Same decomposition as the representative solution below; integrate in Ch 10.

Worked solution (A4 — sup not in the set) #

Set: $A=\{x\in\mathbb{Q}: x^2<2\}\subset\mathbb{R}$.

Every $a\in A$ satisfies $a<\sqrt{2}$ and $a>-\sqrt{2}$. The numbers $\pm\sqrt{2}$ are upper/lower bounds but not in $A$ (irrational).

• Upper bounds: any $o\ge\sqrt{2}$ works; the least upper bound is $\sup A=\sqrt{2}$.
• Lower bounds: symmetrically $\inf A=-\sqrt{2}$.

Check: for any $\varepsilon>0$, $\sqrt{2}-\varepsilon\in A$ (choose rational $q$ with $\sqrt{2}-\varepsilon<q<\sqrt{2}$), so no smaller number is an upper bound.

Worked solution (B1 — partial fractions) #

$\dfrac{x^2+1}{x^3-x}=\dfrac{x^2+1}{x(x-1)(x+1)}$ with $\deg$ numerator $< \deg$ denominator.

Ansatz $\dfrac{A}{x}+\dfrac{B}{x-1}+\dfrac{C}{x+1}$. Clear denominators:

$x=0\Rightarrow 1=-A\Rightarrow A=-1$. $x=1\Rightarrow 2=2B\Rightarrow B=1$. $x=-1\Rightarrow 2=2C\Rightarrow C=1$.

Check: recombine the right-hand side over $x(x-1)(x+1)$ to recover $x^2+1$.

Representative solution (C2) #

Partial fractions: $\dfrac{2x+3}{(x-1)^2(x+2)}=\dfrac{A}{x-1}+\dfrac{B}{(x-1)^2}+\dfrac{C}{x+2}$.

Solving: $A=-\tfrac{1}{9}$, $B=\tfrac{7}{9}$, $C=\tfrac{1}{9}$ (verify by recombination).

Quick decision checklist #

• Negate a statement? → flip quantifiers (§1.1.2).
• Map properties? → definitions in §1.2; composition in §1.2.5.
• Least upper bound / gap in set? → sup/inf (§1.3).
• Estimate $\lvert\cdot\rvert$? → triangle inequality (§1.4).
• Sum $\sum_{k=1}^n(\cdots)$? → telescope (§1.5.1) or known formula (§1.5.3).
• Claim for all $n\in\mathbb{N}$? → induction (§1.6) — first check the claim is true.
• Rational $P/Q$? → §1.7, then Ch 10.

Chapter 2 — Sequences (Folgen) #

Convergence, bounds, Cauchy sequences, and limit computation — the bridge from algebra to analysis.

Story of the chapter: a sequence is an ordered list $a_1,a_2,\ldots$. Analysis asks whether the list settles near a number $L$. Monotone bounded sequences always settle; Cauchy’s criterion lets you prove convergence without knowing $L$ first.

Prerequisites: Chapter 1 (sup/inf, triangle inequality, quantifier negation §1.1).

2.1 Theorems on sequences #

2.1.1 Monotone bounded sequences (Theorem 2.8.3) #

• If $(a_n)$ is monotone increasing and bounded above, then $a_n\to\sup\{a_n:n\in\mathbb{N}\}$.
• If $(a_n)$ is monotone decreasing and bounded below, then $a_n\to\inf\{a_n:n\in\mathbb{N}\}$.

Typical mistake Assuming convergence without proving boundedness (e.g. $a_n=n$ is monotone but diverges).

2.1.2 Cauchy sequences #

$(a_n)$ is a Cauchy sequence iff

In $\mathbb{R}$ (and $\mathbb{C}$): $(a_n)$ converges $\Leftrightarrow$ $(a_n)$ is Cauchy.

Use when You want to prove convergence without guessing the limit.

2.1.3 Contracting sequences (Satz 2.8.10) #

If $0\le c<1$ and $|a_{n+2}-a_{n+1}|\le c|a_{n+1}-a_n|$ for all $n$, then $(a_n)$ converges.

Example (ZSF): $a_{n+1}=\tfrac12(a_n+\sqrt{c})$ with $c\ge 1$, $a_1=\sqrt{c}$ converges to $\sqrt{c}$.

2.1.4 Bolzano–Weierstrass #

Every bounded sequence in $\mathbb{C}$ has at least one accumulation point (subsequence converging to some limit).

Use when Extracting convergent subsequences on compact domains (links Ch 9).

2.1.5 Divergence to $\pm\infty$ #

$a_n\to+\infty$ iff $\forall T\in\mathbb{R}\,\exists N:\ a_n>T$ for all $n\ge N$.

Similarly $a_n\to-\infty$ iff $\forall T$, eventually $a_n<T$.

2.2 Limits #

2.2.1 Important limits (memorize) #

2.2.2 L’Hôpital’s rule #

For $f,g$ differentiable near $x_0$:

when the original limit is an indeterminate form: $\frac00$, $\frac{\infty}{\infty}$, $0\cdot\infty$, $\infty-\infty$, $0^0$, $\infty^0$, $1^\infty$.

Before applying: rewrite $0\cdot\infty$ and $\infty-\infty$ into a quotient.

2.2.3 Special tricks #

• Sandwich: bound $a_n$ between two sequences with the same limit.
• Stolz–Cesàro (exam occasionally): for $\frac{0}{0}$ type $\frac{a_n}{b_n}$ when $b_n\to\infty$.
• Series comparison preview: if terms look like $1/n^p$, compare to Ch 3.

2.3 Method map — limit triage #

1. Plug in — if continuous at the point, done.
2. Standard table (§2.2.1)?
3. Algebra — factor, conjugate, rationalize.
4. Sandwich — especially with trig.
5. L’Hôpital — only after indeterminate form confirmed.
6. Sequence monotone + bounded — find limit via sup/inf.

Typical mistake Using L’Hôpital when the limit is not indeterminate (differentiate incorrectly).

2.9 Sequences — Exercises #

A) Direct #

A1 $\lim_{n\to\infty}\dfrac{3n^2+1}{n^2-n}$.

A2 $\lim_{n\to\infty}\dfrac{\sin n}{n}$.

A3 Show $(a_n)=(1+1/n)^n$ is increasing and bounded above by $3$ (hence convergent; limit is $e$).

B) Multi-step #

B1 $\lim_{x\to 0}\dfrac{e^x-1-x}{x^2}$ (L’Hôpital twice).

B2 Prove $a_{n+1}=\sqrt{2+a_n}$, $a_1=1$ converges and find the limit.

C) Mixed #

C1 $\lim_{n\to\infty}\sqrt{n^2+n}-n$ (hint: multiply by conjugate).

Answers #

A1 $3$. A2 $0$ (sandwich). A3 Standard $e$ proof outline.

B1 $\tfrac12$. B2 Limit $L$ satisfies $L=\sqrt{2+L}\Rightarrow L=2$.

C1 $\dfrac12$.

Chapter 3 — Series (Reihen) #

Infinite sums, convergence tests, and power series — sequences summed into a single number.

Story of the chapter: a series $\sum a_n$ is the limit of partial sums. Absolute convergence is the robust notion (rearrangements safe). Tests tell you which tool fits a given term shape.

Prerequisites: Chapter 2, §1.5 (telescoping sums).

3.1 Theorems on series #

3.1.1 Absolute convergence (Theorem 2.10.7) #

If $\sum_{n=1}^\infty |a_n|$ converges, then $\sum_{n=1}^\infty a_n$ converges and

3.1.2 Leibniz criterion #

If $(a_n)$ is monotone decreasing with $a_n\to 0$, then

converges (conditionally if $\sum|a_n|$ diverges).

3.1.3 Ratio test #

If $a_n\ne 0$ and $|a_{n+1}/a_n|\le c<1$ for all large $n$, then $\sum a_n$ converges absolutely.

If limit $=1$: test inconclusive.

3.1.4 Abel / partial summation (Serie 5 in ZSF) #

If $(a_n)$ decreases to $0$ and partial sums of $(b_n)$ are bounded, then $\sum a_n b_n$ converges.

3.1.5 Root test #

Let $q=\limsup_{n\to\infty}\sqrt[n]{|a_n|}$.

• $q<1$: absolute convergence
• $q>1$: divergence

3.2 Convergence criteria (decision tree) #

Standard facts

3.2.2 Necessary condition #

If $\sum a_n$ converges, then $a_n\to 0$. Contrapositive: if $a_n\not\to 0$, the series diverges.

3.2.3–3.2.4 Majorant / minorant #

Compare $|a_n|$ to a known convergent majorant, or compare positive terms to a divergent minorant.

3.2.5 Integral test #

If $f:[1,\infty)\to\mathbb{R}$ is positive, continuous, decreasing, then $\sum_{n=1}^\infty f(n)$ and $\int_1^\infty f(x)\,dx$ converge or diverge together.

3.3 Power series #

3.3.1 Definition #

is a power series about $x_0$.

3.3.2 Radius of convergence #

(or use ratio test between consecutive coefficients). Converges absolutely for $|x-x_0|<R$; diverges for $|x-x_0|>R$.

Inside the open disk: differentiate and integrate termwise (uniformly on compact subdisks).

3.4 Finite sum formulas #

3.5 Common infinite series (Taylor building blocks) #

Memorize: $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^\alpha$ expansions — used heavily in Ch 6.

Link: full Fourier series on $[-\pi,\pi]$ in Chapter 18.

3.9 Series — Exercises #

A1 Does $\sum \frac{1}{n^{3/2}}$ converge?

A2 $\sum \frac{n!}{3^n}$ (ratio test).

A3 $\sum (-1)^{n+1}/\ln n$ (Leibniz).

B1 Find radius of convergence of $\sum \frac{x^n}{n^2}$.

C1 Decide $\sum \frac{\sin n}{n}$ (Abel / Dirichlet — bounded partial sums of $\sin$).

Answers #

A1 Yes ($p>1$). A2 Diverges (ratio $\to\infty$). A3 Converges conditionally.

B1 $R=1$. C1 Converges (Dirichlet).

Chapter 4 — Functions (Funktionen) #

Continuity, compactness consequences, and the analytic toolkit for limits of functions.

Story of the chapter: a function is continuous if small input changes produce small output changes. On closed bounded intervals, continuous functions attain maxima and minima and take all intermediate values — the theorems behind existence proofs everywhere.

Prerequisites: Chapters 2–3.

4.1 Theorems on functions #

4.1.1 Continuous functions #

$f:D\to\mathbb{R}$ is continuous at $x_0$ iff

Sequential characterization: $f$ continuous at $x_0$ $\Leftrightarrow$ for every sequence $x_n\to x_0$ in $D$, we have $f(x_n)\to f(x_0)$.

4.1.2–4.1.3 Lipschitz and uniform continuity #

Lipschitz: $|f(x)-f(y)|\le L|x-y|$ for some $L\ge 0$.

Uniform continuity: same $\delta$ works for all $x_0$ on the domain.

Lipschitz $\Rightarrow$ uniform continuity $\Rightarrow$ continuity (on bounded domains, not always equivalent).

4.1.5 Intermediate value theorem (IVT) #

If $f:[a,b]\to\mathbb{R}$ is continuous and $f(a)<c<f(b)$ (or reversed), then $\exists\xi\in(a,b): f(\xi)=c$.

4.1.6 Extreme value theorem #

Continuous $f$ on compact $[a,b]$ (or compact $K\subset\mathbb{R}^n$) attains a minimum and maximum on $K$.

4.1.7–4.1.8 Monotonicity and inverse #

Strictly monotone continuous $f$ on an interval has a continuous inverse $f^{-1}$ on $f(I)$.

4.1.9 Mean value theorem (preview) #

See Chapter 6: $\exists c$ with $f'(c)=\dfrac{f(b)-f(a)}{b-a}$.

4.2 Checking continuity #

4.2.1 One dimension #

1. Elementary functions are continuous on their domain.
2. Sums, products, quotients (where defined), compositions preserve continuity.
3. For piecewise definitions: check limits agree at breakpoints.

4.2.2 Several dimensions #

$f:\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$ is continuous at $x_0$ iff

Often proved via sequential characterization or Lipschitz bound on a neighborhood.

4.9 Exercises #

A1 Where is $f(x)=\dfrac{|x|}{x}$ continuous?

B1 Prove $f(x)=\sqrt{x}$ is uniformly continuous on $[0,1]$ but not Lipschitz on $[0,1]$.

C1 Show $x^3-3x+1=0$ has three real roots (IVT on intervals).

Chapter 5 — Sequences of functions (Funktionsfolgen) #

Pointwise vs uniform convergence — when you may swap limit with integral or derivative.

Story of the chapter: $f_n\to f$ pointwise if each fixed $x$ is hit eventually. Uniform convergence adds one speed $N(\varepsilon)$ for all $x$ at once — that extra strength is what makes limit theorems safe.

Prerequisites: Chapters 2–4.

5.1 Convergence modes #

5.1.1 Pointwise convergence #

$f_n:D\to\mathbb{R}$ converges pointwise to $f$ iff $\forall x\in D:\ f_n(x)\to f(x)$.

5.1.2 Uniform convergence #

Equivalent: $\|f_n-f\|_\infty\to 0$ on $D$.

Typical mistake Confusing pointwise with uniform — classic: $f_n(x)=x^n$ on $[0,1]$ converges pointwise but not uniformly.

5.1.3–5.1.5 Uniform limit of continuous functions #

If $f_n$ continuous and $f_n\to f$ uniformly on $D$, then $f$ is continuous.

Cauchy criterion for functions: $(f_n)$ uniformly Cauchy $\Leftrightarrow$ uniformly convergent (on complete codomain).

5.1.5 Normal convergence (series of functions) #

$\sum u_n$ converges normally if $\sum \|u_n\|_\infty<\infty$ $\Rightarrow$ uniform convergence of partial sums.

5.2 Recipe: prove uniform convergence #

1. Estimate $\sup_{x\in D}|f_n(x)-f(x)|$.
2. Show this supremum $\to 0$.
3. Or use Weierstrass $M$-test for series: $|u_n(x)|\le M_n$, $\sum M_n<\infty$.

5.3 Swapping limit with derivative / integral #

Link: improper swapping in §10.1.4.

5.9 Exercises #

A1 Does $f_n(x)=\dfrac{x}{n}$ converge uniformly on $[0,1]$?

B1 $f_n(x)=x^n$ on $[0,1]$: find pointwise limit; uniform?

C1 Use $M$-test on $\sum \dfrac{\sin(nx)}{n^2}$ on $\mathbb{R}$.

Chapter 6 — Derivatives (Ableitungen) #

Rules, mean value theorem, convexity, Taylor polynomials, and Newton’s method.

Story of the chapter: the derivative is the best linear approximation of a function near a point. MVT turns average slope into an instantaneous slope somewhere inside; Taylor refines that to higher order.

Prerequisites: Chapters 4–5, limits from Chapter 2.

6.1 Basics and rules #

Product, quotient, chain rules; derivatives of inverse functions:

6.2 Common derivatives #

Polynomials, exponentials, logs, trig, inverse trig, hyperbolic and inverse hyperbolic (see ZSF table). Link: integral table in Chapter 10 §10.3.1.

6.4 Convex functions #

$f$ convex on $I$ iff $f''(x)\ge 0$ (if twice differentiable) or chord lies above graph.

Inequalities: Jensen-type bounds for exams; tangent line underestimates convex $f$.

6.5 Taylor polynomials #

Integral remainder (Lagrange):

for some $\xi$ between $a$ and $x$.

Product of series: build $f=gh$ from known Taylor factors (ZSF 6.5.3).

6.2.1 Newton’s method #

finds roots when started near a simple zero and $f'$ well-behaved.

6.9 Exercises #

A1 Derivative of $x^x$ ($x>0$).

B1 Taylor of $e^x$ at $0$ to order $3$ with remainder bound on $[-1,1]$.

C1 One Newton step for $f(x)=x^2-2$ from $x_0=1$.

Chapter 7 — Ordinary differential equations (ODEs) #

First-order and linear systems — modeling change and solving with structure.

Story of the chapter: an ODE relates a function to its derivatives. Linearity and constant coefficients let you split into homogeneous + particular parts; systems $\mathbf{x}'=A\mathbf{x}$ use eigenvalues of $A$.

Prerequisites: Chapter 6, Chapter 8 (matrix exponential, optional first pass).

7.1 General workflow #

1. Classify: order, linear/nonlinear, homogeneous/inhomogeneous.
2. Homogeneous solution $y_h$.
3. Particular solution $y_p$ (ansatz, variation of constants, undetermined coefficients).
4. General: $y=y_h+y_p$.
5. Initial conditions fix free constants ($n$ conditions for $n$-th order).

7.2 Homogeneous linear ODEs #

First order: $y'+p(x)y=0$ $\Rightarrow$ $y=Ce^{-\int p}$.

Second order constant coefficients: $ay''+by'+cy=0$. Characteristic equation $ar^2+br+c=0$.

• Distinct real roots $r_1,r_2$: $C_1 e^{r_1 x}+C_2 e^{r_2 x}$.
• Double root $r$: $(C_1+C_2 x)e^{rx}$.
• Complex $\alpha\pm i\beta$: $e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)$ (Euler form: §16.1).

7.3 Particular solutions #

Inhomogeneous $ay''+by'+cy=f(x)$:

• Polynomial / exponential / sine ansatz when $f$ is a finite sum of such terms (adjust if resonance with homogeneous root).
• Variation of parameters when ansatz fails.

Separable: $y'=g(x)h(y)$ $\Rightarrow$ $\int dy/h(y)=\int g(x)\,dx$.

7.4 Matrix exponential #

For diagonalizable $A=PDP^{-1}$: $e^{At}=Pe^{Dt}P^{-1}$.

7.5 Systems #

Convert $n$-th order scalar ODE to first-order system in $\mathbb{R}^n$. Uniqueness: need $n$ independent initial conditions for an $n$-th order equation.

Link: $e^{At}$ and linear maps in Chapter 8.

7.9 Exercises #

A1 Solve $y'=2xy$, $y(0)=1$.

B1 $y''-3y'+2y=e^x$.

C1 System $\mathbf{x}'=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\mathbf{x}$ — periodic solutions (rotation).

Chapter 8 — Differential calculus in $\mathbb{R}^n$ #

Partial derivatives, total differential, Jacobian, chain rule, and implicit functions.

Story of the chapter: in several variables, derivative information lives in the gradient (vector of partials) and Jacobian (matrix). Differentiability means a good linear approximation; the inverse and implicit function theorems say when you can solve locally.

Prerequisites: Chapters 4, 6.

8.1 Concepts #

8.1.1 Partial derivative #

8.1.2 Directional derivative #

(if $f$ differentiable).

8.1.3 Total differentiability #

$f$ differentiable at $a$ iff

where $df(a):\mathbb{R}^n\to\mathbb{R}^m$ is linear (Jacobian matrix).

Hierarchy: $C^1$ $\Rightarrow$ differentiable $\Rightarrow$ partials exist; converse needs continuous partials.

8.1.5–8.1.6 Jacobian and gradient #

8.2 Theorems #

8.2.3 Schwarz #

If $f\in C^2$, then $\dfrac{\partial^2 f}{\partial x_i\partial x_j}=\dfrac{\partial^2 f}{\partial x_j\partial x_i}$.

8.2.4 Inverse function theorem #

If $f\in C^1$ and $df(x_0)$ invertible, then locally $f$ is a $C^1$ diffeomorphism onto its image.

8.2.6 Diffeomorphism checklist #

1. $C^1$ and $\det df(x)\ne 0$ on $U$ $\Rightarrow$ local diffeomorphism.
2. Bijection $U\to V$ $\Rightarrow$ global diffeomorphism (if constructed).

8.2.7 Implicit function theorem #

If $f(x,y)=0$ and $\det D_y f(a,b)\ne 0$, then locally $y=h(x)$ with $f(x,h(x))=0$ and

8.2.8 Existence of extrema on compact sets #

Continuous $f$ on compact $\Omega\subset\mathbb{R}^n$ attains min and max — used in Chapter 9.

8.9 Exercises #

A1 Compute $\nabla f$ for $f(x,y)=x^2y+y^3$.

B1 Implicit derivative: $x^2+y^2=1$, find $dy/dx$.

C1 Verify Schwarz on $f(x,y)=xy^2$.

Chapter 9 — Extrema (Extremwerte) #

Free and constrained optimization in $\mathbb{R}^n$.

Story of the chapter: interior extrema occur where the gradient vanishes (if differentiable). Whether a critical point is min/max/saddle is decided by the Hessian. Constraints add Lagrange multipliers.

Prerequisites: Chapter 8.

9.1 Free extrema #

Critical points: $\nabla f(a)=0$ (or $df(a)=0$).

Hessian test (2D, $C^2$): let $H=\det\begin{pmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{pmatrix}$ at $a$.

• $H>0$, $f_{xx}>0$: local minimum
• $H>0$, $f_{xx}<0$: local maximum
• $H<0$: saddle
• $H=0$: inconclusive

For $n\times n$ symmetric Hessians: eigenvalue test and Sylvester leading minors — §15.7.

Compact domain: compare critical points in interior with boundary values (EVT, Ch 8.2.8).

9.2 Lagrange multipliers #

Optimize $f(x)$ subject to $g(x)=0$. At a regular extremum:

Recipe: form $L=f-\lambda g$, set $\nabla L=0$ and constraint.

9.9 Exercises #

A1 Find and classify critical points of $f(x,y)=x^2+y^2-2x$.

B1 Maximize $xy$ on $x+y=1$, $x,y>0$.

C1 Closest point on $x^2+y^2=1$ to $(2,0)$.

Chapter 10 — Integration #

Limits, definite and improper integrals, integration techniques (substitution, parts, partial fractions, trig substitutions), line integrals, and solids of revolution — with exam-style practice at the end of the chapter.

Story of the chapter: we first define the integral (step functions, Riemann sums), then connect it to antiderivatives (FTC), then learn computational tools (§10.2–10.3) for integrals that are not elementary by inspection.

Prerequisites: limits and continuity (Ch 4), derivatives and FTC intuition (Ch 6), partial fractions (§1.7), uniform convergence when swapping limits (Ch 5 §5.3).

10.1 Definitions and Foundations #

10.1.1 Step Functions #

Definition Let $D = [a,b]$ with $a < b$ . A function $s : D \to \mathbb{R}$ is a step function if there is a partition

such that $s(t) = \sigma_i$ on each $(x_i, x_{i+1})$ .

Integral of a step function

Typical mistake Forgetting this is a signed area (negative values contribute negatively).

10.1.2 Fundamental Theorem of Calculus (FTC) #

Part A (derivative of integral)

if $g$ is continuous.

Part B (evaluation of definite integral)

where $G'(x)=g(x)$ .

Typical mistake Confusing the dummy variable $t$ inside the integral with the outside variable $x$ .

When not to use Part A If $g$ is not continuous on the interval you differentiate across (then $F'$ may fail at jumps).

10.1.3 Riemann Sums #

Here:

• $\Delta x = \frac{b-a}{n}$
• sample point $x_k = a + k\Delta x$ (left endpoints in the formula above; your sum may use a shifted index — that is fine if $x_k$ is consistent)

Identification checklist (do this before integrating)

1. Rewrite the sum as $\displaystyle\sum_{k} f(x_k)\,\Delta x$.
2. Read off $f$, $\Delta x$, and sample points $x_k$.
3. Set $a = x_0$ (or the value at $k=0$ if your indexing starts there).
4. Sanity check: $b-a = n\Delta x$ must hold. If not, $a$ or $b$ is wrong — stop and re-identify.

When to use Exam limits of the form $\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n (\cdots)$ or $\lim_{n\to\infty}\frac{c}{n}\sum (\cdots)$.

When not to use If you cannot match $b-a = n\Delta x$, or the sum is not a Riemann-type sample sum (e.g. missing $\Delta x$, or $x_k$ not affine in $k$).

Typical mistake Missing the factor $\Delta x$ , or forcing $a=0$ when the sample points show a shift (e.g. $1+\frac{2k}{n}$ means $a=1$, not $0$).

10.1.4 Swapping Limit and Integral #

If $f_n \to f$ uniformly on $[a,b]$ , then

Required hypothesis Uniform convergence of $f_n$ to $f$ on $[a,b]$.

When not to use Pointwise convergence alone — there exist sequences with $f_n\to f$ pointwise but $\int f_n \not\to \int f$.

Typical mistake Applying this under only pointwise convergence.

10.1.5 Improper Integrals #

Type 1: Infinite interval

Type 2: Endpoint singularity

(if $g$ is not defined or unbounded at $a$ ).

Two-sided infinite interval

and both integrals must converge.

Typical mistake Treating $\int_{-\infty}^{\infty}$ as a single symmetric limit without checking one-sided convergence.

10.1.6 Inequalities with Integrals #

If $g_1(x)\le g_2(x)$ on $[a,b]$ , then

If $g(x)\ge 0$ , then

If $g\ge 0$ and $[c,d]\subseteq [a,b]$ , then

Typical use Bounding hard integrals by easier comparison functions.

10.1.7 Absolute Value Estimates #

if $|g(t)|\le M$ on the interval.

Interpretation Integral size is controlled by maximum height times interval length.

Typical mistake Using an $M$ that is not valid over the full interval.

10.1.8 Line Integrals (Wegintegrale) #

Scalar field along a curve (arc length form) #

For $f:\mathbb{R}^n\to\mathbb{R}$ and $\gamma:[a,b]\to\mathbb{R}^n$ :

Vector field work integral #

For $\mathbf{v}:\mathbb{R}^n\to\mathbb{R}^n$ and $\gamma:[a,b]\to\mathbb{R}^n$ :

Typical mistake Forgetting $\|\gamma'(t)\|$ in $ds$ -integrals.

10.1.9 Conservative Vector Fields #

Definition Write $\mathbf{v}(x,y)=(P(x,y),Q(x,y))$. The field is conservative on a domain $D$ if there is a potential $\phi$ on $D$ with

When you are allowed to use $\phi(B)-\phi(A)$

1. You have found $\phi$ with $\nabla\phi=\mathbf{v}$ on $D$ (verified by differentiation).
2. Your path from $A$ to $B$ lies entirely in $D$.
3. $D$ is simply connected (or you stay in one branch of $\phi$).

When not to use

• You only “think” a formula looks like an antiderivative — you must check $\partial_x\phi=P$ and $\partial_y\phi=Q$.
• $\dfrac{\partial P}{\partial y}\neq\dfrac{\partial Q}{\partial x}$ on $D$ → not conservative on $D$.
• $D$ has a hole (e.g. plane minus the origin): $\oint_\gamma\mathbf{v}\cdot d\mathbf{r}$ can be $\neq 0$ on a loop around the hole even if $P_y=Q_x$ away from the singularity.

How to test in 2D (simply connected $D$)

Equivalent facts (on simply connected $D$)

• path independence: same endpoints $\Rightarrow$ same line integral;
• $\displaystyle\oint_\gamma \mathbf{v}\cdot d\mathbf{r}=0$ for every closed curve in $D$;
• then $\displaystyle\int_A^B \mathbf{v}\cdot d\mathbf{r}=\phi(B)-\phi(A)$.

Typical mistake Using $\phi(B)-\phi(A)$ without checking $\nabla\phi=\mathbf{v}$, or ignoring holes in the domain.

10.2 Integration Rules #

Computational rules for antiderivatives. FTC (§10.1.2) turns these into definite integrals once you have an antiderivative.

10.2.1 Integration by Parts #

Definite form:

When to use Product integrands where one factor simplifies by differentiation (polynomial, $\ln x$, $\arctan x$, etc.).

When not to use Both factors stay equally messy under differentiation/integration — try substitution or rewrite the integrand first.

Typical mistake Dropped boundary term $\big[uv\big]_a^b$ in definite integrals.

10.2.2 Substitution Rule (u-substitution) #

If $u=h(t)$ , $du=h'(t)\,dt$ , then

Definite form:

When to use When the integrand has an inner function and (up to constants) its derivative is also present:

• chain-rule pattern: $f'(x)\,g(f(x))$
• powers/compositions like $(ax+b)^n$, $e^{h(x)}$, $\sin(h(x))$, $\cos(h(x))$

When not to use No inner function whose derivative (up to a constant) appears in the integrand — try parts, partial fractions, or a trig rewrite.

Typical mistake Changing variable but not changing bounds on definite integrals.

10.2.3 Important Linear Change of Variable #

For $u=at+b$ :

Definite form:

Typical mistake Forgetting the factor $1/a$ .

10.3 Frequent Integrals and Standard Methods #

Method map (read this first)

If two methods seem possible, pick the one that reduces degree or algebraic complexity in one clear step.

10.3.1 Elementary Integrals #

Typical mistake Missing absolute value in logarithms.

10.3.2 Integrals Linked to Inverse Functions #

Typical mistake Confusing $1/(1-x^2)$ with $1/\sqrt{1-x^2}$ .

10.3.3 Power Times Exponential or Trigonometric Function #

Examples:

• $\int x^n e^{ax}\,dx$ - $\int x^n\sin(ax)\,dx$ , $\int x^n\cos(ax)\,dx$

Method Repeated integration by parts:

• differentiate polynomial part,
• integrate exp/trig part,
• repeat until polynomial vanishes.

Typical mistake Stopping before recognizing the recursion pattern.

10.3.4 Exponential-Trigonometric Products #

Standard integrals (template)

Same workflow for both; only the trig factor in the first parts choice changes.

Recommended first choice (cosine case)

Then (schematically, before solving for $I$):

Substitute the second line into the first — the $I$-terms collect on one side.

Algebra pattern (what you are solving)

Here $A(x),B(x)$ are the non-integral terms produced by the two parts steps (products of $e^{ax}$ and trig functions). Do not stop when you still see $\int e^{ax}\cos(bx)\,dx$ or $\int e^{ax}\sin(bx)\,dx$ on the right — finish the second parts, substitute, then solve.

Closed form (check or exam shortcut when $a^2+b^2\neq 0$)

For $a=b=1$: $\displaystyle\int e^x\cos x\,dx=\frac{e^x}{2}(\cos x+\sin x)+C$.

When to use Integrand is $e^{ax}\times$($sin$ or $cos$) — no substitution reduces both factors at once.

When not to use Only one factor is exponential and the other is a polynomial → use repeated parts (§10.3.3). Pure trig products → use identities (§10.3.5).

Typical mistake

• Stopping after one parts step while $\int e^{ax}\sin(bx)\,dx$ (or the cos twin) is still on the right.
• Sign error on $du$ when $u=\cos(bx)$: $du=-b\sin(bx)\,dx$ introduces a plus in front of the second integral.
• Forgetting to move $I$ to the left: $I = \cdots - I$ means $2I=\cdots$, not $I=\cdots$.

10.3.5 Powers of Trigonometric Functions #

Use identities:

General strategy:

• reduce powers,
• rewrite product as sum if needed,
• integrate term-by-term.

Typical mistake Expanding blindly instead of using identities first.

10.3.6 Orthogonal Relations (Fourier Basics) #

Over $[0,2\pi]$ :

Meaning Different frequencies are orthogonal on a full period.

10.3.7 Rational Integrals (Worked-Example Style) #

For integrals like

typical workflow:

1. Split numerator to include derivative of denominator ( $2x+1$ ).
2. Substitution for logarithmic part.
3. Complete square for remaining part:

4. Use arctangent form.

Typical mistake Trying one substitution only; many problems need decomposition first.

10.3.8 Tangent Half-Angle Substitution #

Substitution formulas (set $t=\tan(x/2)$)

Formulas: §15.6.

Useful combinations (after common denominator $1+t^2$)

Workflow

1. Replace $\sin x$, $\cos x$, $dx$ by the formulas above.
2. Simplify — the factor $1+t^2$ in $dx$ often cancels part of the denominator.
3. Integrate the resulting rational function in $t$ (partial fractions / complete the square / log–arctan).
4. Definite integrals: change $x$-bounds to $t$-bounds via $t=\tan(x/2)$ before integrating.
5. Indefinite integrals: at the end, back-substitute $t=\tan(x/2)$ (or use a known $t$-formula).

When to use $\displaystyle\int R(\sin x,\cos x)\,dx$ where $R$ is a rational function (polynomials in $\sin x,\cos x$ divided by each other), and simpler methods (power identities, $\int f(\sin x)\cos x\,dx$, etc.) do not apply.

When not to use

• $\int \sin^n x\cos^k x\,dx$ with an odd power → use §10.3.5 instead.
• $\int f(\sin x)\cos x\,dx$ or $\int f(\cos x)\sin x\,dx$ → simple $u$-sub (§10.2.2).
• Integrands with $\tan x$, $\sec x$ only — sometimes a different $u=\tan x$ or $\sec x$ sub is faster.

Typical mistake

• Forgetting $dx=\dfrac{2}{1+t^2}\,dt$ (missing the factor $2/(1+t^2)$).
• Leaving $x$-bounds on a definite integral after switching to $t$.
• Dropping the domain restriction: $t=\tan(x/2)$ is smooth on $(-\pi,\pi)$; pick the branch that contains your interval.

Reference integral (complete the square on $1+2t-t^2=2-(t-1)^2$)

10.3.9 Solids of Revolution #

Step 0 — Picture the 2D region

You always begin with the same generating region: $\{(x,y): a\le x\le b,\ 0\le y\le f(x)\}$ (or between two curves). The plots below use $f(x)=x^2$ on $[0,1]$ — a curved region (not a triangle), so disk radii and shell heights look different in 2D.

Disk method (slice ⊥ to the x-axis) — rotate around the x-axis

• Slice: a thin vertical strip at position $x$ becomes a disk (pancake) perpendicular to the $x$-axis.
• Radius: distance from the $x$-axis to the curve = $f(x)$ (because the axis is the $x$-axis).
• Thickness: $dx$.
• Volume element: $dV = \pi [\text{radius}]^2\,dx = \pi [f(x)]^2\,dx$.

Disk + 3D (rotation about the $x$-axis): Left — vertical segments are disk radii $r=f(x)$ (slice ⊥ to $x$); notice they shrink toward $x=0$ because $x^2$ is flat there. Right — drag to rotate ($V=\pi/5$ here); red curve = profile in $z=0$.

Shell method (slice ‖ to the y-axis) — rotate around the y-axis

• Slice: a thin vertical strip at position $x$ becomes a cylindrical shell (label on a soup can) with axis parallel to the $y$-axis.
• Radius: horizontal distance from the $y$-axis = $x$.
• Height: $f(x)$ (top of the region at that $x$).
• Thickness: $dx$.
• Volume element: $dV = 2\pi(\text{radius})(\text{height})\,dx = 2\pi x\,f(x)\,dx$.

Shell + 3D (rotation about the $y$-axis): Left — red = shell radius $x$ (horizontal); green = height $f(x)=x^2$ (vertical). At $x=0.75$ the radius is longer than the height — an L-shape, not a single segment like the disk plot. Right — same profile spun about $y$ ($V=\pi/2$).

Which method for which axis?

Same region, different axis → different solid and different volume ($\pi/5$ vs $\pi/2$ for $f(x)=x^2$ on $[0,1]$ below).

Washer method (hole in the middle, axis = $x$):

where $R(x)$ = outer radius, $r(x)$ = inner radius (both measured perpendicular to the $x$-axis). Use when the slice at $x$ is an annulus (disk with a hole), not a solid disk.

Example region: between $r(x)=x^2$ and $R(x)=1$ on $[0,1]$ — rotate about the $x$-axis. Left: outer radius $R(x)$ (red) and inner $r(x)$ (teal) at sample $x$. Right: hollow solid (outer and inner surfaces).

When not to use

• Disk formula with rotation about the y-axis without rewriting $x$ as a function of $y$ — radius is not $f(x)$.
• Shell formula with rotation about the x-axis using $2\pi x f(x)$ — wrong radius (use disks, or shells with thickness $dy$).

Typical mistake Using $f(x)$ as radius when the axis of rotation is the y-axis (radius is $x$, not $y$).

10.4 Integration Exercises (End of Chapter 10) #

Decide first which method fits best — use the method map in §10.3 and the checklist below. Stuck? Riemann limits → §10.1.3; FTC → §10.1.2; bounds after $u$-sub → §10.2.2.

A) Use u-substitution #

1. $\displaystyle \int_0^1 \frac{3x^2}{1+x^3}\,dx$
2. $\displaystyle \int x\,e^{x^2}\,dx$
3. $\displaystyle \int \frac{\cos(\ln x)}{x}\,dx$
4. $\displaystyle \int_0^{\pi/4}\sec^2(2x)\,dx$
5. $\displaystyle \int \frac{dx}{\sqrt{1-4x^2}}$

B) Use integration by parts #

1. $\displaystyle \int x e^x\,dx$
2. $\displaystyle \int \ln x\,dx$
3. $\displaystyle \int x\cos x\,dx$
4. $\displaystyle \int_0^1 x^2\ln(1+x)\,dx$
5. $\displaystyle \int e^x\sin x\,dx$

C) Mixed practice (choose method) #

1. $\displaystyle \int \frac{x^2+1}{x+1}\,dx$
2. $\displaystyle \int_0^2 \frac{x}{(1+x^2)^2}\,dx$
3. $\displaystyle \int \frac{1}{x^2-1}\,dx$
4. $\displaystyle \int \frac{dx}{\sqrt{x^2+4}}$
5. $\displaystyle \int_0^{\pi/2}\frac{dx}{1+\sin x}$

Quick decision checklist #

• Try u-substitution when you see "inside function + derivative of inside" (§10.2.2).
• Try integration by parts for products where differentiating one factor simplifies it (§10.2.1, §10.3.3–10.3.4).
• Try partial fractions for rational functions after polynomial division if needed (§10.3.7).
• Try trig substitution for $\sqrt{a^2-x^2}$, $\sqrt{a^2+x^2}$, $\sqrt{x^2-a^2}$.
• For limits of sums, run the Riemann checklist: $\sum f(x_k)\Delta x$ and $b-a=n\Delta x$ (§10.1.3).
• For improper integrals, split at singularities/infinity and take limits separately (§10.1.5).

Answers (Final Results) #

A) Use u-substitution

1. $\displaystyle \int_0^1 \frac{3x^2}{1+x^3}\,dx=\ln 2$
2. $\displaystyle \int x\,e^{x^2}\,dx=\frac12 e^{x^2}+C$
3. $\displaystyle \int \frac{\cos(\ln x)}{x}\,dx=\sin(\ln x)+C$
4. $\displaystyle \int_0^{\pi/4}\sec^2(2x)\,dx$ diverges (improper integral, $=+\infty$)
5. $\displaystyle \int \frac{dx}{\sqrt{1-4x^2}}=\frac12\arcsin(2x)+C$

B) Use integration by parts

1. $\displaystyle \int x e^x\,dx=e^x(x-1)+C$
2. $\displaystyle \int \ln x\,dx=x\ln x-x+C$
3. $\displaystyle \int x\cos x\,dx=x\sin x+\cos x+C$
4. $\displaystyle \int_0^1 x^2\ln(1+x)\,dx=\frac{2}{3}\ln 2-\frac{5}{18}$
5. $\displaystyle \int e^x\sin x\,dx=\frac12 e^x(\sin x-\cos x)+C$

C) Mixed practice

1. $\displaystyle \int \frac{x^2+1}{x+1}\,dx=\frac{x^2}{2}-x+2\ln|x+1|+C$
2. $\displaystyle \int_0^2 \frac{x}{(1+x^2)^2}\,dx=\frac{2}{5}$
3. $\displaystyle \int \frac{1}{x^2-1}\,dx=\frac12\ln\left|\frac{x-1}{x+1}\right|+C$
4. $\displaystyle \int \frac{dx}{\sqrt{x^2+4}}=\operatorname{arsinh}\!\left(\frac{x}{2}\right)+C$
5. $\displaystyle \int_0^{\pi/2}\frac{dx}{1+\sin x}=1$

Worked Solutions (Step by Step) #

A1 $\displaystyle \int_0^1 \frac{3x^2}{1+x^3}\,dx$

A2 $\displaystyle \int x\,e^{x^2}\,dx$

A3 $\displaystyle \int \frac{\cos(\ln x)}{x}\,dx$

A4 $\displaystyle \int_0^{\pi/4}\sec^2(2x)\,dx$

For the definite integral:

As $b\to\pi/4^-$, $2b\to\pi/2^-$ and $\tan(2b)\to+\infty$, so the integral diverges.

A5 $\displaystyle \int \frac{dx}{\sqrt{1-4x^2}}$

B1 $\displaystyle \int x e^x\,dx$

B2 $\displaystyle \int \ln x\,dx$

B3 $\displaystyle \int x\cos x\,dx$

B4 $\displaystyle \int_0^1 x^2\ln(1+x)\,dx$

Polynomial division:

So

Therefore

B5 $\displaystyle \int e^x\sin x\,dx$ Let

First integration by parts:

Set

Again by parts:

Substitute into $I$:

C1 $\displaystyle \int \frac{x^2+1}{x+1}\,dx$ Polynomial division:

Hence

C2 $\displaystyle \int_0^2 \frac{x}{(1+x^2)^2}\,dx$

C3 $\displaystyle \int \frac{1}{x^2-1}\,dx$ Partial fractions:

So

C4 $\displaystyle \int \frac{dx}{\sqrt{x^2+4}}$ Use $x=2\tan\theta$:

Then

Back-substitute:

so

C5 $\displaystyle \int_0^{\pi/2}\frac{dx}{1+\sin x}$ Rationalize:

Therefore

Take the limit at $\pi/2$:

and at $0$:

Hence

Chapter 11 — Integration Multidimensional $\mathbb{R}^n$ #

Multiple integrals, change of variables (Jacobian), Green's theorem, surface and volume integrals, center of mass, and the divergence theorem (Gauss) — with exam-style practice at the end.

Story of the chapter: in $\mathbb{R}$ we integrated on intervals (Chapter 10). Here the domain is a set $\Omega\subset\mathbb{R}^n$. We approximate $\Omega$ by boxes (Jordan measure), evaluate integrals by Fubini (iterated 1D integrals) on rectangles and normal regions, then change coordinates when the shape is a disk, ball, or cylinder (Jacobian). Line integrals from §10.1.8–10.1.9 connect to Green and Poincaré; flux through a surface leads to Gauss.

Prerequisites from Chapter 10: definite integrals, FTC, line integrals $\int_\gamma \mathbf{v}\cdot d\mathbf{r}$, conservative fields ($\nabla\phi=\mathbf{v}$, $P_y=Q_x$).

Also uses: partial derivatives and Jacobian (Ch 8), parameter surfaces as in Ch 9 / Ch 13.

Theorem ladder (how the pieces fit)

Green is the 2D case of Stokes (Ch 13 §13.0); Gauss is the 3D divergence form of the same “interior cancels, boundary remains” idea.

11.1 Definitions and Foundations #

11.1.1 Boxes, elementary figures, and Jordan measure #

Volume of a box

Riemann integral on a box (same idea as $\mathbb{R}$): partition $Q$ into sub-boxes, use step functions $s\le f\le h$, take infimum of upper sums and supremum of lower sums; if they agree, that value is $\int_Q f\,d\mu$.

Jordan measurable: bounded $\Omega\subset\mathbb{R}^n$ is Jordan measurable iff

equivalently: for every $\varepsilon>0$ there exist elementary figures $E,G$ with $E\subset\Omega\subset G$ and $\mu(G\setminus E)<\varepsilon$.

Typical mistake Confusing "bounded" with "Jordan measurable" — a set can be bounded but have a thick fractal boundary (not Jordan measurable in the classical sense).

11.1.2 Content and notation #

On a Jordan measurable set $\Omega$ and integrable $f$:

All denote the same object. In $\mathbb{R}^2$ we often write $dA$ or $dx\,dy$; in $\mathbb{R}^3$, $dV$.

11.1.3 Fubini's theorem (iterated integrals) #

Theorem (Fubini on a box) Let $Q=[a_1,b_1]\times\cdots\times[a_n,b_n]$ and $f:Q\to\mathbb{R}$ continuous. Then

and any permutation of the integration order gives the same value.

When not to use

• Domain is not a box — first describe $\Omega$ as a normal region (§11.2) or change variables (§11.3).
• $f$ is not continuous on the closure — singularities need splitting (as in improper integrals, §10.1.5).

Typical mistake Wrong bounds when setting up the inner integral (the outer variable must describe the full slice of $\Omega$).

Finer partitions of $Q$ make the boxed approximation fill the volume under the plane; the limit of those sums is $3$.

11.1.4 Domains of class $C^1$, $C^{1}_{\mathrm{pw}}$, $C^k$ #

What you should picture (2D first)

In $\mathbb{R}^2$, near a boundary point $p$ you want a small window where $\Omega$ looks like

with $\psi$ a smooth function (no corner on $\partial\Omega$ inside that window). The formal definition in $\mathbb{R}^n$ is exactly this idea, with the last coordinate playing the role of $y$ and the first $n-1$ coordinates bundled as $x_0$.

Definition (graph over a quader — unpacked)

$\Omega\subset\mathbb{R}^n$ is a $C^1$ domain if for every boundary point $p\in\partial\Omega$ the following local description exists:

Inside $W$, $\Omega$ is exactly the set of points strictly below that graph:

How to read the inequality $c<x_n<\psi(x_0)$

• For each fixed $x_0$ in the base box $Q_0$, the last coordinate $x_n$ may vary in an interval between a floor $c$ and a ceiling $\psi(x_0)$.
• The upper boundary $\partial\Omega$ is the graph $x_n=\psi(x_0)$ (smooth).
• The lower part in this window is flat at $x_n=c$ (that face need not be part of $\partial\Omega$ globally — it is just the bottom of the local box $W$).

At a different boundary point you may need a different coordinate split (e.g. “below a graph in $x$” instead of “below a graph in $y$”). That is allowed: the definition only asks that some smooth graph description works near each $p$.

What is NOT a $C^1$ domain

• A set whose boundary has a reentrant corner (like an inward cusp): near the cusp, no smooth graph $x_n=\psi(x_0)$ describes only one side of $\Omega$.
• “Swiss cheese” with fractal boundary (Jordan measurable can still hold, but not $C^1$).

$C^{1}_{\mathrm{pw}}$ (piecewise $C^1$)

Same local picture, but $\psi$ may be glued from finitely many smooth pieces. A square or rectangle is $C^1_{\mathrm{pw}}$: each edge is a smooth graph in some coordinates; the corners are handled by changing coordinates from edge to edge.

$C^k$

The boundary is described by $\psi\in C^k$ (more derivatives). Stronger regularity for sharper theorems; exams often stop at $C^1$ or $C^1_{\mathrm{pw}}$.

Why it matters

Green's theorem, Gauss's theorem, and the change-of-variables formula are proved for domains whose boundary is well behaved (oriented, piecewise smooth). $C^1$ / $C^1_{\mathrm{pw}}$ is the usual hypothesis so surface integrals $\int_{\partial\Omega}$ and volume integrals $\int_\Omega$ can be related without pathological boundaries.

Typical mistake

• Confusing $C^1$ domain with Jordan measurable: a disk is both; a region with a cusp can still be Jordan measurable but is not $C^1$.
• Applying Green/Gauss on a domain with a hole (annulus) without the inner boundary: $\partial\Omega$ has two closed curves, both oriented with $\Omega$ on the left.

11.2 Integration over Normal Regions #

Type I in $\mathbb{R}^2$ ($x$ outer) #

Then

Type II in $\mathbb{R}^2$ ($y$ outer) #

Then

Method choice

• If vertical slices $\{x=\text{const}\}\cap\Omega$ are always intervals $[f(x),g(x)]$ → Type I.
• If horizontal slices are nicer → Type II.
• Sometimes split $\Omega$ into pieces of different types.

Typical mistake Using $f(x)\le y\le g(x)$ when the region is not a single $x$-strip (e.g. circle needs polar coordinates or two Type-I pieces).

Region for many examples below: between $y=x^2$ and $y=1$ on $x\in[0,1]$.

Picture: walking in $x$ (Type I) #

For Type I you fix $x$ and let $y$ run from the lower graph $f(x)$ to the upper graph $g(x)$. Then you move $x$ from $a$ to $b$ and add up slice areas.

• At $x=0.5$: $f(0.5)=0.25$, $g(0.5)=1$ → strip height $L(x)=g(x)-f(x)=0.75$.
• A thin slab of width $\Delta x$ has area $\approx L(x)\,\Delta x$; shrinking $\Delta x$ makes that rectangle match the true slice (Riemann step toward Fubini).

11.2.1 Normal regions in $\mathbb{R}^3$ (brief) #

Type: $z$ between two graphs over a 2D region $D$ in the $(x,y)$-plane:

Example setup: unit cylinder $x^2+y^2\le 1$, $0\le z\le h$ — integrate $D$ in polar coordinates (§11.3).

11.3 Change of Variables (Jacobian) #

What you should picture (2D)

At a point $\mathbf{u}_0$, the linear map $D\Phi(\mathbf{u}_0)$ sends a tiny square in the $(u,v)$-plane to a parallelogram in the $(x,y)$-plane. Its area is multiplied by $\lvert\det D\Phi(\mathbf{u}_0)\rvert$. Summing over many small squares in $\tilde\Omega$ and taking the limit gives the factor $|\det D\Phi|$ under the integral — polar $r$ is exactly this: the image of a small $r$–$\varphi$ rectangle is approximately a curved patch of area $r\,dr\,d\varphi$.

Coordinates used in §11.3 (symbol table)

Here $\Phi$ sends parameter space (where the integral is easy) to physical space (where the set $\Omega$ lives). The factor $|\det D\Phi|$ (or $r$, $r^2\sin\vartheta$ from the cheat sheet) is the local area/volume scaling.

Theorem (transformation rule, $\mathbb{R}^2$) Let $\Omega\subset\mathbb{R}^2$ be Jordan measurable, $\Phi:U\to\mathbb{R}^2$ a $C^1$ diffeomorphism on a neighborhood of $\bar\Omega$ with $\tilde\Omega=\Phi^{-1}(\Omega)\subset U$. If $f$ is integrable on $\Omega$, then

$3$D: replace $dA$ by $dV$, $|\det D\Phi|$ is the $3\times 3$ Jacobian determinant.

Typical mistake Forgetting $|\det D\Phi|$ or using the wrong sign — always take absolute value for unsigned area/volume integrals.

When not to use

• $\Phi$ is not one-to-one on the domain (e.g. polar at the origin) — restrict to a set where it is bijective, or use symmetry.
• Jacobian vanishes somewhere (fold) — transformation invalid there.

11.3.1 Jacobian cheat sheet #

Use $J=\lvert\det D\Phi\rvert$ and $dA$ or $dV = J\,du\,dv$ (new coordinates on the right).

Convention note: ETH scripts sometimes use $\vartheta$ = polar angle from $z$-axis and $\varphi$ = azimuth; always match your lecture's symbol table.

Standard volumes (memorize)

11.3.2 Polar coordinates ($\mathbb{R}^2$) #

Meaning of $r$ and $\varphi$

• $r\ge 0$ = distance from the origin to $P=(x,y)$.
• $\varphi$ = angle from the positive $x$-axis to the ray $OP$ (counterclockwise).

On that ray, trigonometry in the right triangle gives

Point and triangle

Drag $\varphi$ and $r$ — watch $P$ move. Left: $(x,y)$ with red $r=|OP|$, blue along the $x$-axis ($x=r\cos\varphi$), and green vertical at that $x$ ($y=r\sin\varphi$) — a right triangle on the coordinate axes, not a second pair of axes. Right: the same point in the $(r,\varphi)$ plane.

Why the Jacobian factor is $r$

In the $(r,\varphi)$-plane, a small rectangle $dr\,d\varphi$ maps to a curved patch in $(x,y)$. The arc length in the angular direction is about $r\,d\varphi$ (not $d\varphi$), so area scales by $r$:

Jacobian patch (interactive). Left: the image in $(x,y)$ of a small rectangle in $(r,\varphi)$. Right: that same rectangle in the parameter plane — its area is always $\Delta r\,\Delta\varphi$ (fix $\Delta r$, $\Delta\varphi$ and it does not change).

1. Vary $r_0$, keep $\Delta r$, $\Delta\varphi$ fixed. The rectangle on the right stays the same size; the curved patch on the left grows or shrinks. Closer to the origin ($r_0$ small) the patch has less area; farther out ($r_0$ large) it has more. The caption below the plot reports $\Delta r\,\Delta\varphi$ and an estimate $r_0\,\Delta r\,\Delta\varphi$ for the patch area; their ratio stays $\approx r_0$. That is why $dA\approx r\,dr\,d\varphi$: the same $(dr,d\varphi)$ cell carries more area at larger $r$.

2. Shrink $\Delta r$, $\Delta\varphi$. Both panels show a smaller cell; the ratio in the caption still tracks $r_0$ (the linear approximation improves as the cell gets tiny).

Annulus domain $r_1\le r\le r_2$, $0\le\varphi\le 2\pi$: washer in $(x,y)$ matches the rectangle in $(r,\varphi)$. Use the r and φ sliders to see the same point in both planes — inner/outer circles correspond to $r=r_1$ and $r=r_2$; the full $\varphi$ edge is one full turn.

11.3.3 Cylindrical and spherical ($\mathbb{R}^3$) #

Compact $dA$, $dV$ factors: §15.12.

Cylindrical $(r,\varphi,z)$

Same horizontal map as polar; the $z$-coordinate is unchanged. A vertical column above a point in the disk is a line segment in $z$.

Cylinder $x^2+y^2\le R^2$, $0\le z\le h$:

Spherical $(r,\vartheta,\varphi)$

• $r\ge 0$ = distance from origin.
• $\vartheta\in[0,\pi]$ = polar angle from the $+z$-axis (colatitude; $\vartheta=0$ is north pole, $\vartheta=\pi$ is south pole).
• $\varphi\in[0,2\pi)$ = azimuth in the $xy$-plane (same role as polar $\varphi$).

Project $P$ onto the $xy$-plane: horizontal radius is $r\sin\vartheta$, then polar formulas give

The volume element is $r^2\sin\vartheta\,dr\,d\vartheta\,d\varphi$ (factor $r^2\sin\vartheta$ from stretching in $r$, $\vartheta$, and $\varphi$).

Jacobian patch (interactive). Left (3D): the curved shell in $(x,y,z)$ for the parameter brick $[r_0,r_0+\Delta r]\times[\vartheta_0,\vartheta_0+\Delta\vartheta]\times[\varphi_0,\varphi_0+\Delta\varphi]$ — inner cap at $r_0$, outer at $r_0+\Delta r$, four side faces, plus a faint reference sphere at $r=1.2$; axes span the ball but the default view is zoomed on the patch (drag to rotate, scroll out to see the full sphere). Right (3D): the same cell as an ordinary axis-aligned box in the full $(r,\vartheta,\varphi)$ domain — the wireframe brick moves when you change $r_0$, $\vartheta_0$, or $\varphi_0$; $\Delta r$, $\Delta\vartheta$, $\Delta\varphi$ are its edge lengths.

1. Increase $\Delta r$. Left: shell thickens between inner and outer caps. Right: box grows in the $r$ direction. Metrics: $\Delta r\,\Delta\vartheta\,\Delta\varphi$ increases.

2. Vary $\vartheta_0$, keep all $\Delta$ fixed. Right box stays the same shape; left shell changes size ($\sin\vartheta$ factor). Ratio $r_0^2\sin\vartheta_0$ in the caption tracks $\vartheta_0$.

3. Shrink all $\Delta$. Both cells shrink; ratio $\approx r_0^2\sin\vartheta_0$ stays stable (linearization improves).

Ball $x^2+y^2+z^2\le R^2$:

11.4 Green's Theorem and Poincaré #

Increase $n$ below: the partition is grid lines only (not a carpet of cell arrows). CCW arrows sit on the outer boundary $\partial\Omega$ only; one highlighted interior edge shows the same segment traced twice in opposite directions (cell A: $+$, cell B: $-$) — those contributions cancel. Only the outer rim survives when you sum all cells.

Green's theorem Let $\Omega\subset\mathbb{R}^2$ be a bounded $C^{1}_{\mathrm{pw}}$ domain, $\partial\Omega$ oriented so $\Omega$ lies on the left as you walk along $\partial\Omega$. Let $P,Q\in C^1(\bar\Omega)$. Then

Typical mistake

• Treating Green (always a conversion between $\iint_\Omega$ and $\oint_{\partial\Omega}$) like Poincaré (only when $\mathbf{v}=\nabla\phi$ on a simply connected domain).
• Using Poincaré on a region with a hole after checking only $P_y=Q_x$ — circulation around the hole can still be $\neq 0$ (see §11.4.1).

Common form (area): with $P=0$, $Q=x$:

Left panel (swirl plot below): $\displaystyle\oint_{\partial\Omega}(P\,dx+Q\,dy)$ is an integral along the boundary curve. Gold arrows on $\partial\Omega$ show which way you walk (CCW; $\Omega$ on the left) — that orientation was always there; earlier they were orange and easy to confuse with $P$. Along each tiny step $d\mathbf{r}=(dx,dy)$ you add $P\,dx+Q\,dy$ using the field at that boundary point (muted arrows show $\mathbf{v}=(P,Q)$ in $\Omega$). At the probe $(x_0,y_0)$ only: orange horizontal $=P$ (coefficient of $dx$), red vertical $=Q$ (coefficient of $dy$), dark diagonal $=\mathbf{v}$ — a decomposition at one interior point, not the walk direction. On the circular arc, gold ticks can look parallel to $\mathbf{v}=(-y,x)$ by geometry; they are still traversal, not $P$. Right panel: $Q_x-P_y$ mosaic (here constant $2$).

Connection to §10.1.9 (conservative fields)

If $\mathbf{v}=(P,Q)=\nabla\phi$, then $P_y=Q_x$ (no swirl). Green’s left side is $0$, so $\oint_{\partial\Omega}\mathbf{v}\cdot d\mathbf{r}=0$ on any closed boundary in a region without holes — consistent with $\int_A^B \mathbf{v}\cdot d\mathbf{r}=\phi(B)-\phi(A)$ on open curves only.

11.4.1 Poincaré lemma (2D) #

On a simply connected bounded $C^{1}_{\mathrm{pw}}$ domain $\Omega$, for $\mathbf{v}=(P,Q)\in C^1(\bar\Omega;\mathbb{R}^2)$, the following are equivalent:

1. $\mathbf{v}$ is conservative ($\mathbf{v}=\nabla\phi$ on $\Omega$).
2. $\displaystyle\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$ on $\Omega$.
3. $\displaystyle\oint_\gamma \mathbf{v}\cdot d\mathbf{r}=0$ for every closed curve $\gamma$ in $\Omega$.

When not to use

• Domain has a hole (annulus): $P_y=Q_x$ may hold but $\oint$ around the hole can be $\neq 0$ — no global potential on all of $\Omega$.

The rotation field below is tangential on circles; right panel: $Q_x-P_y\approx 0$ (flat tiles) even though $P_y=Q_x$ away from the origin. Caption: $\oint_{\partial\Omega}$ can still be nonzero on the outer boundary — the hole blocks a global potential (Poincaré fails).

11.5 Surface Integrals #

Parametrization $\Phi:B\to\mathbb{R}^3$, $B\subset\mathbb{R}^2$.

Scalar (first kind) #

Vector / flux (second kind) #

Orientation: $d\mathbf{o}=\mathbf{n}\,dS$ with $\mathbf{n}$ unit normal; the cross product $\partial_u\Phi\times\partial_v\Phi$ gives the correct orientation if $\mathbf{n}$ points the way you want (outward for closed surfaces enclosing volume).

Recipe (ZSF-style)

1. Parametrize surface $\Phi(u,v)$.
2. Compute $\Phi_u$, $\Phi_v$, then $\mathbf{N}=\Phi_u\times\Phi_v$.
3. Check normal direction (flip $(u,v)$ order if needed).
4. Substitute into integral.

Typical mistake Using $\|\Phi_u\|\,\|\Phi_v\|$ instead of $\|\Phi_u\times\Phi_v\|$ — wrong except for orthogonal parameter lines in special cases.

11.6 Volume Integrals #

Compute by

1. Fubini on a normal region in $\mathbb{R}^3$ (§11.2.1): fix $(x,y)\in D$, integrate $z$ between graphs $\varphi_1\le z\le\varphi_2$, then integrate over $D$.
2. Substitution when $V$ is a ball, cylinder, or sector — polar / cylindrical / spherical (§11.3) with the Jacobian factor.

Do not mix up surface integrals (§11.5, 2D parameter domain) with volume integrals (solid $V\subset\mathbb{R}^3$).

Typical mistake Setting up a volume integral without a clear order of integration — draw the projection onto the $(x,y)$-plane first, then fix $z$-bounds.

11.7 Center of Mass (Schwerpunkt) #

Formulas

For a solid $K\subset\mathbb{R}^n$ with density $\rho$ (often $\rho\equiv 1$):

Symmetry shortcuts

• If $K$ is symmetric under $x\mapsto -x$ and $\rho$ is even in $x$, then $s_x=0$ without integrating. Similarly for $y,z$.
• Uniform density on a body symmetric about the $z$-axis $\Rightarrow$ $s_x=s_y=0$; only $s_z$ may need a calculation.

Typical mistake

• Using $\iiint x_i\,\rho\,dV$ as the center of mass — forgets dividing by $M$.
• Ignoring $\rho$ in $M$ while integrating $x_i\rho$ in the numerator (inconsistent density).

11.8 Gauss's Theorem (Divergence Theorem) #

Definition (divergence)

In $\mathbb{R}^3$: $\mathrm{div}\,(v_1,v_2,v_3)=\partial_x v_1+\partial_y v_2+\partial_z v_3$.

Gauss Let $V\subset\mathbb{R}^3$ be bounded with $C^{1}_{\mathrm{pw}}$ boundary $\partial V$, outward unit normal $\mathbf{n}$. For $\mathbf{v}\in C^1(\bar V;\mathbb{R}^3)$:

When to use

• Closed surface + field smooth inside → try div and volume integral (often easier than parametrizing $\partial V$).
• Compare with §11.5 flux: same integral, Gauss is the shortcut when $V$ is solid and closed.

Synthesis: operator identities, Stokes, and when to pick Green vs Stokes vs Gauss — Ch 13 §13.3, §13.5.

Typical mistake Wrong orientation (inward normal) — pick outward $\mathbf{n}$ for the standard Gauss formula.

Useful identities

• $\mathrm{div}(\nabla\times\mathbf{K})=0$.
• $\mathrm{div}(f\mathbf{K})=\nabla f\cdot\mathbf{K}+f\,\mathrm{div}\,\mathbf{K}$.

11.9 Integration in $\mathbb{R}^n$ — Exercises #

Use §11.1–11.8 and Chapter 10 line integrals where needed.

A) Fubini and normal regions #

1. $\displaystyle\int_0^1\int_0^{x^2} (x+y)\,dy\,dx$
2. Area of $\{(x,y):0\le y\le 1,\ y\le x\le 2-y\}$
3. $\displaystyle\iiint_V z\,dV$ where $V$ is the box $[0,1]^3$

B) Jacobian (polar / cylindrical / spherical) #

4. $\displaystyle\iint_{x^2+y^2\le 4} (x^2+y^2)\,dA$
5. $\displaystyle\int_0^{2\pi}\int_0^1\int_0^2 r\,dz\,dr\,d\varphi$ (cylinder setup — evaluate)
6. Volume of $\{(x,y,z):x^2+y^2+z^2\le 9,\ z\ge 0\}$ (hemisphere)

C) Green and conservative fields #

7. $\displaystyle\oint_\gamma (y\,dx+2x\,dy)$ where $\gamma$ is the unit circle counterclockwise
8. Is $\mathbf{v}=(2xy+y^2,\ x^2+2xy)$ conservative on $\mathbb{R}^2$? If yes, find a potential.

D) Surface integrals #

9. $\displaystyle\iint_A z\,dS$ where $A$ is the plane $z=1$ over $x^2+y^2\le 1$
10. Flux of $\mathbf{K}=(0,0,1)$ through $z=1$, $x^2+y^2\le 1$ (upward normal)

E) Volume and center of mass #

11. Mass of unit cube $[0,1]^3$ with $\rho(x,y,z)=xyz$
12. $s_z$ for uniform solid cylinder $x^2+y^2\le 1$, $0\le z\le 2$

F) Gauss #

13. $\displaystyle\iint_{\partial V}\mathbf{v}\cdot\mathbf{n}\,dS$ for $\mathbf{v}=(x^2,y,z)$, $V=$ unit cube $[0,1]^3$
14. $\mathrm{div}$ of $(yz,xz,xy)$ and flux through sphere radius $1$ centered at origin

Quick decision checklist #

• Box or normal region? → Fubini with explicit bounds (§11.1.3, §11.2).
• Disk, annulus, ball, cylinder? → polar / cylindrical / spherical (§11.3).
• $\oint P\,dx+Q\,dy$ on closed curve in $\mathbb{R}^2$? → Green or check $P_y=Q_x$ (§11.4, §10.1.9).
• Flux through a surface? → parametrize (§11.5); if closed surface of a solid → Gauss (§11.8).
• Mass / center of mass? → $\iiint \rho\,dV$ and symmetry (§11.6–11.7).

Answers (final results) #

A) 1. $\frac{7}{30}$ — 2. $1$ — 3. $\frac{1}{2}$

B) 4. $8\pi$ — 5. $4\pi$ — 6. $18\pi$

C) 7. $\pi$ — 8. yes, $\phi(x,y)=x^2y+xy^2+C$

D) 9. $\pi$ — 10. $\pi$

E) 11. $\frac{1}{8}$ — 12. $s_z=1$

F) 13. $3$ — 14. $\mathrm{div}=0$, flux $0$

Worked solutions (selected) #

A1

B4 Polar: $\int_0^{2\pi}\int_0^2 r^2\cdot r\,dr\,d\varphi=2\pi\cdot\frac{16}{4}=8\pi$.

C7 Green: $\partial Q/\partial x-\partial P/\partial y=2-1=1$, area of unit disk $=\pi$.

C8 $P=2xy+y^2$, $Q=x^2+2xy$. Then $P_y=2x+2y$, $Q_x=2x+2y$ — equal on $\mathbb{R}^2$, so conservative. Find $\phi$:

So $\phi(x,y)=x^2y+xy^2+C$. Verify: $\phi_x=2xy+y^2=P$, $\phi_y=x^2+2xy=Q$.

D10 $\mathbf{K}=(0,0,1)$ on $z=1$, $x^2+y^2\le 1$, upward. $\Phi=(x,y,1)$, $\Phi_x\times\Phi_y=(0,0,1)$, $\mathbf{K}\cdot(\Phi_x\times\Phi_y)=1$, flux $=\iint_{x^2+y^2\le 1}1\,dx\,dy=\pi$.

F13 $\mathrm{div}\,\mathbf{v}=2x+1+1=2x+2$,

F14 $\partial_x(yz)+\partial_y(xz)+\partial_z(xy)=0+0+0$, flux $0$ by Gauss.

Chapter 12 — Potential fields (Potenzialfelder) #

A potential field is a vector field that is a gradient: an “antiderivative” in $\mathbb{R}^2$ or $\mathbb{R}^3$, so line integrals reduce to endpoint differences.

Story of the chapter: if $\mathbf{v}=\nabla\phi$ on $\Omega$, work along any path in $\Omega$ depends only on endpoints, not on the route. Finding $\phi$ is an integration problem. Topology (simply connected vs holes) decides whether “curl-free” ($P_y=Q_x$) actually implies a global $\phi$.

Prerequisites: §10.1.9 (exam recipe), §11.4.1 (holes), §11.4 connection to conservative fields.

12.0 What a potential field buys you #

Definition: $\phi\in C^1(\Omega)$ is a potential of $\mathbf{v}=(P,Q)$ on $\Omega$ if $\mathbf{v}=\nabla\phi$, i.e. $\phi_x=P$ and $\phi_y=Q$ on $\Omega$.

$n$-dimensional integrability: on simply connected $\Omega\subset\mathbb{R}^n$, $\mathbf{v}$ is a gradient iff $\partial v_i/\partial x_j=\partial v_j/\partial x_i$ for all $i\ne j$.

12.1 Method A — partial integration (see §10.1.9) #

Full step table: §10.1.9 (test $P_y=Q_x$ $\Rightarrow$ $\phi=\int P\,dx+h(y)$ $\Rightarrow$ match $Q$ $\Rightarrow$ verify).

12.2 Method B — path integral along a broken line #

If Method A is awkward, fix a base point $(x_0,y_0)$ and integrate along horizontal then vertical (or the reverse, staying in $\Omega$):

12.3 When $P_y=Q_x$ is not enough (holes) #

Simply connected $\Omega$: every closed curve in $\Omega$ can be shrunk to a point inside $\Omega$ (no holes). For $\mathbf{v}\in C^1(\Omega)$,

With a hole (e.g. annulus around the origin): $P_y=Q_x$ can still hold on $\Omega$, but a loop around the hole may have $\oint\neq 0$, so there is no single $\phi$ on all of $\Omega$ with $\nabla\phi=\mathbf{v}$.

12.4 Quick decision checklist (exam) #

1. Closed curve in $\mathbb{R}^2$? $\Rightarrow$ try $\oint(P\,dx+Q\,dy)$ or Green (§11.4); use $\phi(B)-\phi(A)$ only if you know $\mathbf{v}$ is conservative on the region.
2. Open curve, simply connected region? $\Rightarrow$ test $P_y=Q_x$ $\Rightarrow$ find $\phi$ (§10.1.9, §12.1, or §12.2) $\Rightarrow$ verify $\Rightarrow$ $\int_A^B\mathbf{v}\cdot d\mathbf{r}=\phi(B)-\phi(A)$.
3. Hole / annulus? $\Rightarrow$ do not assume a global $\phi$ from $P_y=Q_x$ alone; see §11.4.1.

12.5 Exercises and answers #

A1 On the annulus $\{0.5\le r\le 1.5\}$, is $\mathbf{v}=\bigl(-\tfrac{y}{x^2+y^2},\,\tfrac{x}{x^2+y^2}\bigr)$ a gradient?

A2 On $\mathbb{R}^2$, is $\mathbf{v}=(2x,2y)$ a gradient?

B1 Find a potential for $\mathbf{v}=(e^x\cos y,\,-e^x\sin y)$ on $\mathbb{R}^2$.

B2 Using a potential, evaluate $\displaystyle\int_{(0,0)}^{(1,1)}(2xy+y^2,\,x^2+2xy)\cdot d\mathbf{r}$.

Answers

• A1: No. $P_y=Q_x$ on the annulus, but $\oint=2\pi\neq 0$ on a circle around the hole (§12.3, Example B).
• A2: Yes. $\phi=x^2+y^2+C$ (§12.3, Example A).
• B1: $\phi(x,y)=e^x\cos y+C$ (full path-integral solution in §12.2).
• B2: $\phi=x^2y+xy^2+C$ gives $\phi(1,1)-\phi(0,0)=2$ (§12.1, Example A).

Chapter 13 — Vector analysis (Vektoranalysis) #

grad, div, rot, operator identities, and a map of Green / Stokes / Gauss — when to use which.

Story of the chapter: Chapter 11 shows how to compute flux and volume integrals. Here you learn what $\mathrm{grad}$, $\mathrm{div}$, and $\mathrm{rot}$ mean, and how the three big theorems are one ladder: interior derivative $\leftrightarrow$ boundary integral. Stokes in 3D is the piece Green does not cover alone.

Prerequisites: Chapter 11 (§11.4 Green, §11.5 flux, §11.8 Gauss); Chapter 12 ($\mathbf{v}=\nabla\phi$, curl-free fields).

13.0 FTC ladder (Green / Stokes / Gauss) #

Same idea as the Ch 11 theorem ladder: partition the domain into small pieces; interior contributions cancel; only the boundary remains.

Reading the table: Green is Stokes in the plane (normal $\mathbf{k}$, curl has only a $z$-component). Gauss is the divergence version: sources inside $V$ $\Leftrightarrow$ net outward flux through $\partial V$. Full statements and proofs: §11.4, §13.3, §11.8.

13.1 Operators (grad, div, rot) #

For a scalar $f$ and vector field $\mathbf{F}=(F_1,F_2,F_3)$ in $\mathbb{R}^3$:

(Gradient in $\mathbb{R}^n$: §8.1.5–8.1.6 — Jacobian and gradient.)

Curl mnemonic (ZSF 13.2.2): $\mathrm{rot}\,\mathbf{F}=\nabla\times\mathbf{F}$ is the symbolic determinant

13.2 Identities (ZSF 13.2.3) #

Exam use

• $\mathrm{div}(f\mathbf{K})$ — split a flux integral when $\mathbf{v}=f\mathbf{K}$ (see §11.8 product rule).
• $\mathrm{div}(f\,\mathrm{rot}\,\mathbf{K})$ — when flux involves $f$ times a rotation field; reduces to $\nabla f\cdot\mathrm{rot}\,\mathbf{K}$.
• $\mathrm{rot}(\nabla f)=0$ — every gradient field has no swirl; if $\mathbf{v}=\nabla\phi$ then $\mathrm{rot}\,\mathbf{v}=\mathbf{0}$ (Chapter 12). On simply connected $\Omega$, $\mathrm{rot}\,\mathbf{v}=\mathbf{0}$ $\Rightarrow$ path independence (Stokes).
• $\mathrm{div}(\mathrm{rot}\,\mathbf{K})=0$ — swirl fields have no net sources; Gauss on a region with only $\mathrm{rot}\,\mathbf{K}$ often gives flux $0$.
• In $\mathbb{R}^2$, $P_y=Q_x$ for $\mathbf{v}=(P,Q,0)$ $\Leftrightarrow$ $(\mathrm{rot}\,\mathbf{v})_z=0$.
• $\mathrm{div}(\nabla f)=\Delta f$ — Poisson / Laplace context (scalar).

13.3 Theorem map (Green / Stokes / Gauss) #

Green as 2D Stokes: for $\mathbf{v}=(P,Q,0)$ on a plane region, $Q_x-P_y=(\mathrm{rot}\,\mathbf{v})_z$.

Stokes (classical): oriented surface $S$ with piecewise smooth boundary $\partial S$. Choose $\mathbf{n}$ on $S$; traverse $\partial S$ counterclockwise when viewed from the tip of $\mathbf{n}$ (right-hand rule).

Gauss: outward unit normal $\mathbf{n}$ on closed $\partial V$. Inward normal flips the sign.

Partial boundary (ZSF 13.2.6): Gauss needs a closed $\partial V$. If you only want flux through one piece $S\subset\partial V$ (e.g. a hemisphere), close $V$ mentally, compute $\iiint_V \mathrm{div}\,\mathbf{v}\,dV$, then subtract flux through the other faces of $\partial V\setminus S$ (§13.6).

Area via Green (ZSF 13.2.4 / 13.2.8): $\mathrm{Area}(D)=\displaystyle\iint_D 1\,dA$. With $\mathbf{v}=(0,x,0)$, $(\mathrm{rot}\,\mathbf{v})_z=1$, so $\mathrm{Area}(D)=\oint_{\partial D}\mathbf{v}\cdot d\mathbf{r}$ (CCW). Full steps: §13.4b.

13.4 Stokes — worked example #

13.4b Green — area via line integral (ZSF 13.2.8) #

13.5 Flux: when to parametrize vs Gauss vs Stokes #

Rules

1. Closed $\partial V$ + smooth $\mathbf{v}$ on $\bar V$ → compute $\mathrm{div}\,\mathbf{v}$, use §11.8.
2. Open surface (hemisphere, graph, patch) → usually §11.5: $\boldsymbol{\Phi}(u,v)$, $\mathbf{N}=\boldsymbol{\Phi}_u\times\boldsymbol{\Phi}_v$, integrate $\mathbf{v}\cdot\mathbf{N}\,du\,dv$.
3. Circulation on $\partial S$ with easy $\mathrm{rot}\,\mathbf{F}$ → Stokes (§13.4).
4. Need $\phi(B)-\phi(A)$? → Chapter 12, not this chapter.

Scalar surface area (no vector field): §11.5 Example A (hemisphere area).

13.6 Example — hemisphere flux (exercise C1) #

13.7 Exam checklist #

1. Closed curve in $\mathbb{R}^2$? → Green or $P_y=Q_x$ / potential (§11.4, Ch 12).
2. Area of plane region? → Green with $\mathbf{v}$ such that $(\mathrm{rot}\,\mathbf{v})_z=1$ (§13.4b).
3. Closed surface of a solid? → $\mathrm{div}\,\mathbf{v}$ + Gauss (§11.8).
4. Flux through one face only? → Gauss on full $V$, subtract other faces (§13.3, §13.6).
5. Surface with boundary + circulation? → $\mathrm{rot}\,\mathbf{F}$ + Stokes (§13.4).
6. Open surface, flux only? → Kochrezept (§13.5) or Gauss trick from (4).
7. Compute $\mathrm{rot}$ or $\mathrm{div}$ first — wrong operator $\Rightarrow$ wrong theorem.
8. Orientation: outward $\mathbf{n}$ for Gauss; right-hand rule for Stokes/Green (§13.3).

13.9 Exercises and answers #

A1 Compute $\mathrm{rot}\,(x,y,z)$.

A2 Compute $\mathrm{rot}\,(y,-x,0)$.

B1 Verify $\mathrm{div}(\mathrm{rot}\,\mathbf{v})=0$ for $\mathbf{v}\in C^2$.

B2 On the unit circle in the $xy$-plane, evaluate $\displaystyle\oint(-y\,dx+x\,dy)$ using Stokes on the unit disk.

C1 Flux of $\mathbf{v}=(0,0,z)$ through upper hemisphere $x^2+y^2+z^2=1$, $z\ge 0$ (outward).

C2 Area of $\{x^2+y^2\le 4\}$ using Green with $\mathbf{v}=(0,x,0)$.

C3 Simplify $\mathrm{div}\bigl(x\,\mathrm{rot}(y,-x,0)\bigr)$ at a point (no integral).

Answers

• A1: $\mathrm{rot}\,(x,y,z)=\mathbf{0}$.
• A2: $(0,0,-2)$.
• B1: Write $\mathrm{rot}\,\mathbf{v}=(R_1,R_2,R_3)$; then $\mathrm{div}(\mathrm{rot}\,\mathbf{v})=\partial_x R_1+\partial_y R_2+\partial_z R_3=0$ (equality of mixed partials / Schwarz).
• B2: $2\pi$ (§13.4).
• C1: $\dfrac{2\pi}{3}$ (§13.6).
• C2: $4\pi$ (disk radius $2$: $\oint x\,dy=\int_0^{2\pi}4\cos^2 t\,dt=4\pi$, or $\pi r^2=4\pi$).
• C3: $\mathrm{rot}(y,-x,0)=(0,0,-2)$; $\mathrm{div}(x\,\mathrm{rot}\,\mathbf{K})=\nabla x\cdot\mathrm{rot}\,\mathbf{K}=1\cdot(-2)=-2$ (constant).

Chapter 14 — Topology (Topologie) #

Open sets, closure, compactness, and simply connected domains — why integration theorems have hypotheses on $\Omega$.

Story of the chapter: analysis is not only formulas. Theorems say “$\Omega$ open”, “$K$ compact”, “simply connected” because the shape of the domain matters. This chapter names those shapes and links them to Ch 11 (integrals), Ch 12 (potentials / holes), and Ch 4 (extrema on compact sets).

Prerequisites: Chapter 4 (continuity); helpful after §12.3 (hole vs no hole).

14.0 Why topology for analysis #

14.1 Open and closed sets #

For $x_0\in\mathbb{R}^d$ and $r>0$, the open ball is

Definitions

• $x_0$ is an inner point of $\Omega\subset\mathbb{R}^d$ if $\exists r>0$ with $B_r(x_0)\subseteq\Omega$.
• $\Omega$ is open if every point of $\Omega$ is an inner point.
• $A\subset\mathbb{R}^d$ is closed if $A^c=\mathbb{R}^d\setminus A$ is open.

Useful rules (ZSF 14.1.3–14.1.5)

• $\emptyset$ and $\mathbb{R}^d$ are open (and closed).
• Finite intersections of open sets are open; arbitrary unions of open sets are open.
• Finite unions of closed sets are closed; arbitrary intersections of closed sets are closed.

Sequential test: $A$ is closed iff whenever $x_k\in A$ and $x_k\to x$, then $x\in A$.

14.2 Interior, closure, boundary #

For $A\subset\mathbb{R}^d$:

• Interior $\mathring{A}$ = union of all open sets contained in $A$ (largest open subset).
• Closure $\overline{A}$ = smallest closed set containing $A$; equivalently $\overline{A}=\{x:\ \exists (x_k)\subset A,\ x_k\to x\}$.
• Boundary $\partial A=\overline{A}\setminus\mathring{A}$.

Relations: $\mathring{A}\subseteq A\subseteq\overline{A}$ and $\overline{A}=\mathring{A}\sqcup\partial A$ (disjoint union).

Boundary point: $x\in\partial A$ iff every ball $B_r(x)$ meets both $A$ and $A^c$.

14.3 Compactness in $\mathbb{R}^n$ #

Bounded: $K\subset\mathbb{R}^d$ is bounded if $K\subseteq B_R(0)$ for some $R$.

Compact (Heine–Borel in $\mathbb{R}^d$): for $K\subset\mathbb{R}^d$,

Sequential characterization: $K$ compact $\Leftrightarrow$ every sequence in $K$ has a convergent subsequence with limit in $K$.

Why exams care

• Extreme value theorem: continuous $f$ on compact $K$ attains $\min$ and $\max$ on $K$ (§4.1.6).
• Continuous image: if $f$ continuous and $K$ compact, then $f(K)$ is compact.
• Uniform limits: on compact $K$, uniform limit of continuous functions is continuous (Ch 5).

14.4 Continuity and open sets #

For $f:\Omega\to\mathbb{R}^m$ with $\Omega\subset\mathbb{R}^d$ open, the following are equivalent (ZSF 14.3):

1. $f$ is continuous (sequential / $\varepsilon$–$\delta$ — Ch 4).
2. For every open $V\subset\mathbb{R}^m$, the preimage $f^{-1}(V)=\{x\in\Omega:\ f(x)\in V\}$ is open in $\Omega$.
3. For every closed $B\subset\mathbb{R}^m$, $f^{-1}(B)$ is closed in $\Omega$.

Exam use: prove a set is closed by writing it as $f^{-1}(B)$ for continuous $f$ and closed $B$ (as in §14.3 Step 1 with $g_1(x,y)=x$, $g_3(x,y)=3x+y$ and $B=(-\infty,0]$ or $[3,\infty)$).

Norms (one line): equivalent norms on $\mathbb{R}^d$ define the same open sets (ZSF 14.2.2) — so “open” does not depend on whether you use the Euclidean norm or $\|\cdot\|_\infty$.

14.5 Relatively open and closed (ZSF 14.1.10) #

Let $X\subset\mathbb{R}^n$. A set $A\subset X$ is relatively open in $X$ if $A=B\cap X$ for some open $B\subset\mathbb{R}^n$. Similarly relatively closed in $X$.

Example: $X=[0,1)$ and $A=[0,\tfrac12)$. Then $A=(-\tfrac12,\tfrac12)\cap X$ with $(-\tfrac12,\tfrac12)$ open in $\mathbb{R}$, so $A$ is relatively open in $X$ even though $A$ is not open in $\mathbb{R}$.

Why it appears: continuity on a domain $X$ that is not open in $\mathbb{R}^n$ (e.g. a curve or a square with boundary) uses relative topology.

14.6 Connected and simply connected #

Connected: $U\subset\mathbb{R}^d$ cannot be split into two disjoint non-empty sets that are both relatively open in $U$.

Simply connected (exam level in $\mathbb{R}^2$): every closed curve in $U$ can be shrunk continuously to a point without leaving $U$ (no holes). Same hypothesis in Cauchy’s theorem (§16.3).

14.7 Jordan measurable (link Ch 11) #

Definition (§11.1.1): bounded $\Omega\subset\mathbb{R}^n$ is Jordan measurable iff its boundary $\partial\Omega$ has $(n-1)$-dimensional measure zero.

Intuition: you can approximate $\Omega$ by unions of boxes from inside and outside; “nice” regions for Riemann integration in $\mathbb{R}^n$.

Examples

• Disk, rectangle, ball: Jordan measurable (smooth boundary).
• $\mathbb{Q}\cap[0,1]^2$: bounded but not Jordan measurable (boundary too thick).

Do not confuse: Jordan measurable $\not=$ simply connected; Jordan measurable $\not=$ $C^1$ domain (§11.2, §11.4).

14.8 Exam checklist #

1. Open or closed? Inner points / complement open / sequence test (§14.1).
2. Interior, closure, boundary? Use definitions or ZSF table (§14.2).
3. Compact in $\mathbb{R}^n$? Closed + bounded (§14.3).
4. Prove closed: preimage of closed set under continuous map (§14.4).
5. Potential / Green: simply connected or hole? (§14.6, Ch 12).
6. Set up integral: Jordan measurable? (§14.7, Ch 11).

14.9 Exercises and answers #

A1 Is $(0,1)$ open in $\mathbb{R}$? Is $[0,1]$ closed?

A2 For $A=[0,1]$, find $\mathring{A}$, $\overline{A}$, $\partial A$.

B1 Give a subset of $\mathbb{R}$ that is bounded but not compact.

B2 Is $D=\{(x,y):x\ge 0,\ y\ge 0,\ 3x+y\le 3\}$ compact in $\mathbb{R}^2$?

C1 Is the annulus $\{0.5\le x^2+y^2\le 1.5\}$ simply connected?

C2 Is $\mathbb{Q}\cap[0,1]$ Jordan measurable?

Answers

• A1: $(0,1)$ is open; $[0,1]$ is closed (§14.1).
• A2: $\mathring{A}=(0,1)$, $\overline{A}=[0,1]$, $\partial A=\{0,1\}$ (§14.2).
• B1: $(0,1)$ — bounded in $\mathbb{R}$ but not closed, hence not compact.
• B2: Yes — closed and bounded (§14.3).
• C1: No — a loop around the inner circle cannot shrink to a point inside the annulus (§14.6, §12.3).
• C2: No — boundary too large / not a “nice” Jordan region (§14.7).

Chapter 15 — Reference formulas #

Compact reference for exam lookup — standard identities, roots, and $dV$ factors in one place.

Use when: during a calculation you need a formula you do not remember.

15.1 Mitternachtsformel #

For $a\neq 0$:

15.2 $pq$-formula #

For $x^2+px+q=0$ (leading coefficient $1$):

15.3 Trigonometric values #

15.4 Addition theorems and powers #

Core

Product-to-sum

Inverse trig (domain restrictions as in course)

Power reductions

Phase shifts

15.5 Hyperbolic identities #

Same bridge as Euler’s formula in §16.1.

Inverses

15.6 Tangent half-angle #

With $t=\tan\frac{x}{2}$:

15.7 Matrices — inverse and definiteness #

$2\times 2$ inverse ($\det A=ad-bc\neq 0$):

Definiteness (symmetric $H$)

15.8 Diagonalization recipe #

For $A\in\mathbb{R}^{n\times n}$:

1. Characteristic polynomial: $\det(A-\lambda I)=0$ $\Rightarrow$ eigenvalues $\gamma_1,\ldots,\gamma_n$.
2. For each $\gamma_k$, solve $(A-\gamma_k I)\mathbf{v}=\mathbf{0}$ for eigenvector $\mathbf{v}_k$.
3. $S=[\mathbf{v}_1\ \cdots\ \mathbf{v}_n]$, $D=\mathrm{diag}(\gamma_1,\ldots,\gamma_n)$.
4. If $S$ invertible: $A=SDS^{-1}$.

15.9 Binomial and polynomial tricks #

Factoring

15.10 $\max$ as formula #

(Combination of continuous functions $\Rightarrow$ continuous.)

15.11 Geometric sets #

15.12 Coordinate parametrizations #

Graph surface $z=f(x,y)$ over $D$:

15.13 Quick integral primitives #

Standard antiderivatives:

15.14 Quick lookup exercises #

A1 Solve $x^2-5x+6=0$ via $pq$. A2 Exact value of $\sin(75°)$ using $\sin(45°+30°)$.

B1 Is $\begin{pmatrix}2&1\\1&2\end{pmatrix}$ positive definite? B2 Volume of a cone radius $3$, height $4$.

Answers

• A1: $p=-5$, $q=6$ $\Rightarrow$ $x=2$ or $x=3$.
• A2: $\sin 75°=\sin 45°\cos 30°+\cos 45°\sin 30°=\frac{\sqrt6+\sqrt2}{4}$.
• B1: Yes — leading minors $2>0$, $\det=3>0$ (§15.7).
• B2: $V=\frac{1}{3}\pi\cdot 9\cdot 4=12\pi$ (§15.11).

Chapter 16 — Complex numbers ($\mathbb{C}$) #

Algebra, polar form, holomorphic functions, and contour integrals at exam level.

Story of the chapter: $\mathbb{C}$ completes the algebra of polynomials and encodes rotation and oscillation through $e^{i\theta}$. For analysis exams you need three layers: arithmetic/polar, holomorphic + Cauchy–Riemann, and contour integrals / residues.

Prerequisites: Chapter 2 (limits), Chapter 6, Chapter 8 (partials for CR).

16.1 Algebra, polar form, and Euler #

Basics

$z=x+iy$, $\overline{z}=x-iy$, $\Re z=x$, $\Im z=y$, $i^2=-1$.

Division

In polar form: divide moduli, subtract arguments.

Polar / Euler

Multiplication: $|z_1 z_2|=|z_1||z_2|$, $\arg(z_1 z_2)=\arg z_1+\arg z_2$ mod $2\pi$.

De Moivre

$n$th roots of $z\neq 0$: there are exactly $n$ distinct roots

Link: complex characteristic roots $\alpha\pm i\beta$ in §7.2 give $e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)$ — the same rotation as $e^{(\alpha+i\beta)x}$.

Typical mistakes

• Treating $\arg(z_1 z_2)=\arg z_1+\arg z_2$ as an exact equality without mod $2\pi$.
• Forgetting all $n$ roots of $z^n=w$ (e.g. $z^2=-1$ gives $\pm i$).

16.2 Holomorphic functions and Cauchy–Riemann #

$f:\Omega\subset\mathbb{C}\to\mathbb{C}$, write $f=u+iv$ with $u,v:\Omega\to\mathbb{R}$.

Complex derivative at $z_0$:

(limit must exist for every complex $h\to 0$, not only along axes).

Holomorphic on $\Omega$ if $f'$ exists at every point of $\Omega$.

Cauchy–Riemann (CR): if $u,v\in C^1$, then $f$ holomorphic on $\Omega$ $\Rightarrow$

If $u,v\in C^1$ and CR hold on $\Omega$, then $f$ is holomorphic on $\Omega$.

Harmonic: if $f$ holomorphic, then $u$ and $v$ are harmonic ($\Delta u=0$, $\Delta v=0$).

Examples

• $f(z)=z^2=(x^2-y^2)+2ixy$: $u_x=2x=v_y$, $u_y=-2y=-v_x$ — holomorphic.
• $f(x,y)=x$ only (no $y$ in imaginary part): $u_x=1$, $v_y=0$ — not holomorphic.

Typical mistake Checking $u_x=v_y$ but forgetting $u_y=-v_x$ (or testing CR only along $x$ / $y$ directions).

16.3 Cauchy theorems and contour setup #

Contour $\gamma:[a,b]\to\mathbb{C}$ piecewise $C^1$:

Cauchy (simply connected): if $f$ is holomorphic on a simply connected open $\Omega$ and $\gamma$ is a closed contour in $\Omega$, then

Hypothesis: “simply connected” means no holes — see §14.6. On an annulus, $f$ can be holomorphic with $\oint\neq 0$.

Path independence: on simply connected $\Omega$, $\int_\gamma f\,dz$ depends only on the endpoints of $\gamma$.

Typical mistakes

• Applying Cauchy on a domain with a hole without checking simply connected.
• Parametrizing $\gamma(t)$ but omitting $\gamma'(t)$ in $dz$.

16.4 Cauchy integral formula #

If $f$ is holomorphic on $|z-z_0|<R$ and $|w-z_0|<r<R$:

Derivatives ($n\ge 0$):

Use when: the integrand is $\dfrac{g(z)}{(z-a)^{n+1}}$ with $g$ holomorphic inside the circle — read off $f^{(n)}(a)$ from $g$.

16.5 Residues and contour integrals #

Residue at an isolated singularity $a$:

Simple pole at $a$:

Standard templates

• $\operatorname{Res}\bigl(\dfrac{1}{z-a},a\bigr)=1$.
• For $\dfrac{g(z)}{z-a}$ with $g$ holomorphic and $g(a)\neq 0$: $\operatorname{Res}(f,a)=g(a)$.

Pole of order $m\ge 2$ at $a$:

Residue theorem

Typical mistakes

• Counting a pole outside $\gamma$.
• Using the simple-pole formula on a double pole.

Out of scope here: full Laurent-series chapter; essential singularities (only: do not blindly use simple-pole rules).

16.9 Exercises #

A1 Compute $(1+i)^8$ using De Moivre.

A2 Verify CR for $f(z)=z^2$ written as $u(x,y)+iv(x,y)$.

A3 $\displaystyle\oint_{|z|=3}\frac{1}{z(z-2)}\,dz$.

B1 Polar form of $-1+i\sqrt{3}$.

B2 Is $e^z$ holomorphic on $\mathbb{C}$? (Use $e^{x+iy}=e^x(\cos y+i\sin y)$.)

B3 $\displaystyle\oint_{|z|=2}\frac{z}{z^2+1}\,dz$.

Answers

• A1: $|1+i|=\sqrt2$, $\arg(1+i)=\pi/4$ $\Rightarrow$ $(1+i)^8=(\sqrt2)^8 e^{i\cdot 8\cdot\pi/4}=16 e^{i2\pi}=16$.
• A2: $u=x^2-y^2$, $v=2xy$; $u_x=2x=v_y$, $u_y=-2y=-v_x$ — holomorphic.
• A3: Partial fractions $\dfrac{1}{z(z-2)}=\dfrac{1}{2(z-2)}-\dfrac{1}{2z}$. Residues $+\tfrac12$ at $z=2$ and $-\tfrac12$ at $z=0$; both inside $|z|=3$; sum $=0$.
• B1: $|z|=2$, $\arg z=2\pi/3$ $\Rightarrow$ $z=2e^{i2\pi/3}$.
• B2: Yes — $e^z$ is entire (holomorphic on all of $\mathbb{C}$).
• B3: Poles at $\pm i$ inside $|z|=2$. At $z=i$: $\operatorname{Res}=\dfrac{i}{2i}=\tfrac12$; at $z=-i$: $\dfrac{-i}{-2i}=\tfrac12$; integral $=2\pi i$.

Chapter 17 — Additional integrals (Zusätzliche Integrale) #

Supplement to Chapter 10 — special forms from the ZSF / Dirren appendix.

Use when: the integrand matches one of the templates below and §10.2–10.3 did not immediately apply.

17.1 Substitution templates #

17.2 Worked example #

(see §10.3 rationalization family).

Chapter 18 — Fourier series (Fourierreihe) #

Orthogonal expansion of periodic functions — full theory beyond §10.3.6.

Story of the chapter: periodic signals repeat; Fourier series splits them into sines and cosines (eigenmodes of translation). Coefficients are projections using orthogonality.

Prerequisites: Chapter 3, §10.3.6.

18.1 Orthogonality on $[-\pi,\pi]$ #

18.2 Fourier coefficients #

For $2\pi$-periodic $f$ integrable on $[-\pi,\pi]$:

Even $f$: only cosines; odd $f$: only sines.

18.3 Convergence (exam level) #

For piecewise $C^1$ periodic $f$, series converges to $\tfrac12(f(x^+)+f(x^-))$ at jumps.

Typical mistake Differentiating termwise without checking uniform convergence of derivative series.

18.4 Parseval (energy) #

18.9 Exercises #

A1 Fourier coefficients of $f(x)=x$ on $(-\pi,\pi)$ (odd → sine series).

B1 Square wave $f(x)=\mathrm{sgn}(\sin x)$ — sketch partial sums.